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Chapter 6 Hyperbolic dynamics and riemannian geometry
CHAPTER
6
Hyperbolic Dynamics and Riemannian Geometry Gerhard Knieper Fakultiit f~r Mathematik, Ruhr- Universitiit Bochum, NA 05/33, 44780 Bochum, Germany E-mail: gknieper@ math. ruhr-uni-bochum, de
Contents I. Basics on Riemannian geometry and geodesic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
455
I.I. Levi-Civita connection and the linearization of the geodesic flow . . . . . . . . . . . . . . . . . . .
455
1.2. Jacobi equation
459
..............................................
1.3. Geodesic Anosov flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Geometry of manifl~lds of nonpositive curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Hadamard manilolds
...........................................
473 476 476
2.2. Rigidity of the sphere at inlinity in negative curvature . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Busemann functions and the horospherical foliations . . . . . . . . . . . . . . . . . . . . . . . . . .
479 485
2.4. Volume growth and entropy formulas
487
..................................
2.5. Poincard series and Buscmann densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Horospherical foliations
............................................
3. i. Ergodic theory of the loliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Regularity of the foliation and rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Entropy and rigidity
..............................................
489 491 491 492 495
4. I. Entropy comparison and rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
495
4.2. Regularity of topological entropy
5(X)
....................................
4.3. Minimal entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
502
4.4. Spectral rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
509
4.5. Rigidity of the marked length spectrum
515
.................................
5. Ergodic properties of weakly hyperbolic spaces 5. I. Definition of rank and rank rigidity
...............................
...................................
515 515
5.2. Uniqueness of the Busemann density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Volume growth and growth rate of regular closed geodesics . . . . . . . . . . . . . . . . . . . . . .
518 520
5.4. Construction of an ergodic measure of maximal entropy . . . . . . . . . . . . . . . . . . . . . . . .
528
5.5. The uniqueness of the measure of maximal entropy
532
5.6. Growth rate of singular closed geodesics 6. Appendix
..........................
................................
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6. !. An intrinsic proof of Pestov's identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. I A Edited by B. Hasselblatt and A. Katok 9 2002 Elsevier Science B.V. All rights reserved 453
536 537 537 543 543
454
G. Knieper
Abstract In this survey we will focus on recent results on the dynamics of geodesic flows on compact manifolds of nonpositive curvature with hyperbolic or at least weakly hyperbolic behavior. Furthermore, we will discuss rigidity theorems on compact negatively curved manifolds like entropy rigidity, the minimal entropy problem, rigidity of stable and unstable foliations and the infinitesimal spectral rigidity theorem. In many cases we will provide complete proofs and not only statements of theorems. Some of the results are new and did not appear somewhere else. For instance, we give a new upper bound for the number of closed geodesics in rank 1 manifolds and a new entropy comparison result. Moreover, we will give an intrinsic proof of a formula of Weitzenb6ck type on the tangent bundle (Pestov's identity). However, we will not discuss geodesic flows on manifolds with positive curvature. The reason is that until now there is no example of such a flow with a hyperbolic set of positive Liouville measure. Since we do not leave the geometric setting, we do not discuss the interesting subject of general Euler-Lagrange flows. Many of those topics are covered in the recent book by G. Paternain [76] which is in some sense complementary to this survey. Furthermore, we do not cover geodesic metric spaces as for instance Gromov hyperbolic spaces or the Alexandrov geometry of nonpositively curved metric spaces. This theory is treated in many text books and surveys as for instance [6,16,42].
Hyperbolic dynamics and Riemannian geometry.
455
1. Basics on Riemannian geometry and geodesic flows 1.1. Levi-Civita connection a n d the linearization o f the geodesic f l o w As a general reference for the basic theory on Riemannian geometry we refer to the following sources [21,53,61,80]. Let M be a smooth (usually C ~ ) manifold, denote by T M ~ I , ~ M T p M its tangent bundle and let n" 9T M ~ M be the canonical projection. A family g - - ( , ) -- {gp}p~M
of inner products gp on T p M depending differentiably on p is called a Riemannian metric. The tangent space 7',, T M of T M has for every v 6 T M a natural linear subspace tangent to the fibers which is, therefore, generated by smooth curves X : ( - s , + e ) --+ T M , X (0) = v such that ~ l , = 0 7 r X ( s ) - 0. This space is called the vertical space V,, and is given by I/',, = ker dTr,,. The Riemannian metric g induces a natural horizontal space H,, via the Levi-Civita connection D. This connection is uniquely determined by the fact that it is torsion free and that the metric g is parallel. The horizontal subspace H,, of T,, T M is generated by parallel /) vector fields, i.e., by smooth curves X" ( - s , + s ) --~ T M , X(O) -- v such that ~ X ( s ) - - 0 , where ~17 denotes the covariant derivative along the curve y (s) "-- Jr o X (s). Hence, if we define the connection map C,, : 7",,T M --~ Tn,,M to be
C,, ~
.s=()
X(s)
-~,,.:o
'
the horizontal space H,, is equal to the kernel of C,,. Since H,, O V , , - {0} and dim H,, = dim V , , - d i m M , we obtain T~,TM -- H,, 9 V,,. Therefore, the map (drr,,, C,,)" T , , T M ---> T~,,M • T~,,M given by ~ ~ (dn',,(~), C,,(~)) is a linear isomorphism. Consequently, H,,~{(u',())Iu'ETrr,,M}
and
V , , ~ { ( 0 , u ' ) I u , ETzr,,M}.
The geodesics c in M are smooth curves with parallel velocity vector, i.e., /7i. -- 0. They are also characterized by their energy minimizing property. The Riemannian metric induces a distance function d on M. In the sequel we will assume that the metric space (M, d) is complete. By the theorem of H o p f - R i n o w this is equivalent to the fact that M is geodesically complete. Denote by c,, :F~ ---> M the geodesic with initial condition (',,(0) = v. Then the geodesic flow 4)t : T M ---> T M is given by 4)t (v) = b,,(t). Therefore, d dt
4)' (v) ~ t =()
(dl ~
Jrr (v), 0
)
- (v, 0),
t =()
i.e., the infinitesimal generator of the geodesic flow is X ( ; ( v ) -- (v, 0). The main reason for introducing the above decomposition of T T M in horizontal and vertical space is that it
G. Knieper
456
provides a convenient way for computing the differential of the geodesic flow using Jacobi fields. Jacobi fields are solutions of the linear equation
J"(t) + R ( J ( t ) , ckt (v))ckt (v) = 0 , where
J: =
(1.1)
D j and R(X, Y ) Z denotes the curvature tensor.
LEMMA 1.1. The differential
D e t (v)'Trr,,M x Tjr,,M--+ Tjrr
x Trrr
is given by DckI (v)(x, y) = (J(t), J'(t)), where J(t) is the Jacobi field along cv with J (0) -- x, J:(O) -- y. PROOF. Let X ' ( - e , X (0) = v
+e) --+ T M be a smooth curve such that and
D
d ds
X ( s ) = y.
Jr X ( s ) = x , s =0
s =0
Then, using the chain rule and the definition of the horizontal and vertical spaces, we obtain ?) D e I (v)(x, y)"--
,=or
8 o X (s)
D
Jr o r
-
~ ,=or ~X (s)).
s =()
Since (s,t) -+ rr o 4; o X(s) defines a geodesic variation of the geodesic 7r o r the horizontal component J ( t ) " - - ~il [s=()7r o t~t X(s) is a Jacobi field and it's covariant derivative D]
D 0
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~r x(s) - ~
J ' ( t ) = t-~ ~-~s s----{)
~r
x (,,,) -
D
s =()
r s =()
l-I
is the vertical component. There is a natural l-form (,9 on T M defined by CO,,(~) = (v, dn,, (~)). Using the decomposition of ~ introduced above, we obtain
(x, y ) ~
T,,TM in the horizontal and vertical part,
fO,,(x, y) -- (v, x). LEMMA 1.2. The differential d69 is a symplectic form given by dO,,((xl, Yl), (x2, Y2)) - ( y l , x2) - (y2, xl).
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Hence, the infinitesimal Hamiltonian vector field submanifolds of constant We will mainly study manifold of energy 1/2) SM-
{v E Y g
generator X G ( V ) = (v, 0) of the geodesic flow is the of the kinetic energy. In particular, the flow leaves the energy invariant. the geodesic flow on the unit tangent bundle (the sub-
l llvll - 1},
where Ilvll- (v, v)!/2. The tangent space T v S M is given by {(x, y) l x , y E T~rr
y 2_ v},
since for each smooth curve X ' ( - e ,
+ e ) ---> S M with X(0) -- v it follows that
0 - ~1,-0(x(s), x ( s ) ) - (~l,=0x(s), v). (b) Denote by ECb(v) = s p a n { X , ; ( v ) } -- {(Xv, O) l~. ~ R} the 1-dimensional space tangent to the geodesic flow at v E S M , and by E 4' the associated vector bundle. The interesting part of the dynamics occurs transversal to the flow lines. The next lemma shows that there is a canonical invariant bundle transversal to the flow. LEMMA 1.4. The orthogonal complement
N ( v ) "-- E ~ ( v ) Ain T~,SM, with respect to the Sasaki metric, defines a bundle N invariant under the linearization o f the geodesic flow. Furthermore, N is a symplectic subbundle, i.e., the symplectic f o r m restricted to N is nondegenerate. PROOF. Since N ( v ) -- {(x,y) I x 2_ v, y 2_ v} the nondegeneracy follows from the definition of w; the invariance follows from L e m m a 1.3. [-1 Assume that dim M = n -+- l and, therefore, the dimension of the bundle N is equal to 2n. In order to describe the linearization of the geodesic flow restricted to N it is convenient to trivialize N along the flow lines 4)t (v) in the following way. Choose a parallel orthonormal frame El (t) . . . . . E , ( t ) orthogonal to 4~I (v) and consider for t 6 R the linear map p, " R" • R" ----~ N(dp' (v)) given by
p t ( x , y) - -
yi Ei(t)
x i Ei(t), i=1
,
Hyperbolic dynamics and Riemannian geometry
459
where X i and yi are the coordinates of x and y. Denote by w0 and go the standard symplectic and Euclidean structure on R" • R n, i.e., 090((Xl,Yl), (x2, Y 2 ) ) - - y l
" X2 - - X l " y 2
and
go((xl, yl), (x2, y2)) -- xl "X2 -k- Yl " Y2, where x 9 y -- Y]'i'= ! x i y i is the standard scalar product on II{". For each t ~ R the map p, transforms the symplectic structure oJ and the Sasaki metric gs restricted to N ( r (v)) to the standard symplectic and Euclidean structure on IR" x IR". Furthermore, the linearization D e ' (V)lN(v):N(t~) ~ N ( r t (19)) can be described in the following way. The curvature tensor induces for each t a symmetric endomorphism R ( t ) : r (v) • --~ r (v) • given by R ( t ) w = R(w, r (v))r (v). Associated to the parallel orthonormal frame, we obtain the symmetric matrix representation R(t)i.i = ( R ( E i ( t ) , Ct(v))r (v), E j ( t ) ) which we also denote by R(t). With respect to this framing, the Jacobi equation (1.1) translates into the second order ODE on R" (1.2) which we call Jacobi equation as well. Now, Lemma l.l immediately implies. LEMMA 1.5. If J(t) -- ( j l ( t ) . . . . . J " ( t ) ) is a solution of the Jacobi equation
)'(t) + R ( t ) J ( t ) --0,
(I.2)
then (J(t), j ( t ) ) -- p~-i
De' (v)p{}(J(O),
)(0)).
In the next section, we will study the Jacobi equation in the framework of linear symplectic geometry on a Euclidean vector space. For the basic terminology in symplectic geometry we refer to [26].
1.2. Jacobi equation Let (E, ( , ) ) be a vector space with scalar product ( , ) . Let End(E) denote the set of endomorphisms and let Sym(E) denote the set of symmetric endomorphisms on E. If R" It{ -+ Sym(E) is a smooth C ~ - m a p (curvature operator), the linear differential equation ~176
J ( t ) -k- R ( t ) J
-- 0
is called the Jacobi equation and its solutions J" ~ ~ B" ]R --+ End(E) is a solution of the matrix equation
B(t) + R ( t ) B ( t ) - - 0 ,
E are called Jacobi fields. If
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Hyperbolic dynamics and Riemannian geometry
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REMARK. The Wronskian and the symplectic structure are related in the following way. If A, B :[a, b] ---> E n d ( E ) are C ! -curves, then
(F2(A(,), B(t))x, y ) - co((A(t)x, .4(t)x), (B(t)y, /}(,)y)) for all x, y e E. LEMMA 2.2. Let X, Y : R ~ E n d ( E ) be Jacobi tensors. Then
F2(X(t), Y(t)) = const for all t e IR. PROOF. Since X ( t ) x and Y(t)y are Jacobi fields, the proof follows from the remark above. [--I Important notions in symplectic geometry which are also useful in our context are isotropic and Lagrangian subspaces. DEFINITION 2.3. A linear subspace L C E x E is called isotropic, iff w(~e, 71) : 0 fl)r all ~, ~1e L. L is called Lagrangian, iff L is isotropic and dim L = dim E.
REMARK. Since c,) is nondegenerate, the dimension of isotropic spaces is always less than or equal to dim E. LEMMA 2.4. Let A, B : E ~
E be endomorphisms o r E and
L -- {(Ax, B x ) l x e E I .
Then L is isotreq~ic if and only i['A r B = B ~A. If additionally x ~ (Ax, Bx) is in.jective, L is Lagrangian. In particuhtr, the graph L - - { ( x , Ax)[ x e E I
is l_ztgnmgian ~['and only i['A is symnletrir PROOF. The proof follows from o)((Ax, Bx), (Ay, By)) -- (Bx, Ay) - (Ax, By) --(A r Bx, y ) - (B' Ax, y).
I---1
REMARK. Note that L = {(Ax, Bx) Ix r E} is a graph if and only if A is nonsingular. DEFINITION 2.5. Let R :Ii~ ~
S y m ( E ) be a smooth map. A family of Lagrangians Lt C E x E, t E [a, b], with Lr = 7,J'I.,,L, is called a Lagrangian deformation of L,, induced by R.
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Hyperbolic dynamics and Riemannian geometry
Integration yields
A(t)-- B(t) fa I (BTB)-I(s)BT(a)ds. This implies that A is nonsingular and, therefore, there are no conjugate points.
C1
The variational aspects of the Jacobi equation is, as we will see, revealed by the index form. DEFINITION 2.8. Let R : R ---> S y m ( E ) be a curvature operator and V = {X : [a, b] ---> E I X piecewise differentiable} be the vector space of continuous and piecewise differentiable curves in E. The symmetric bilinear form I := ll.,t,l: E x E - + R
with
f
II,,,,,I(X, Y ) =
h
(X(t), ~ ' ( t ) ) -
(R(t)X, Y)dt
is called the index form. If we want to specify the curvature operator, we write I k or I k
I.,/'1"
REMARK. Let X, Y 6 V be differentiable on each interval of the subdivision a = to < tt < 9.- < tk -- b. Then, using integration by parts, we obtain
k I(X, Y)-- Z ( , Y , Y) i--I
Ii li t I
c Ii'(J( + R(t)X, V)dt. J,
(2.2)
This implies that X is a Jacobi field if and only if I(X, Y) = 0 for all Y :[a, b] --+ E with Y(a) = Y(b) = 0. Since
I(X + s Y , X + s Y ) = 21(X, Y),
s=() X is a Jacobi field if and only if it is a critical point of the action I(X) : = I(X, X) on the space of vector fields with fixed end points. This is not surprising, since the Jacobi equation is nothing else than the E u l e r - L a g r a n g e equation for the time dependent Lagrangian 1
L(t,x, 3')- -~((Y, Y) - (R(t)x,x)) on[a,b]•
E x E.
464
G. Knieper
LEMMA 2.9 (Index L e m m a ) . Assume that the Jacobi equation has no conjugate points on [a, b]. Then I[a,hl is positive definite on the subspace
V~
{X 9 V I X ( a ) -
X(b)--O},
i.e.,
I(X,X)>0
for all X 9 V ~ X r O.
PROOF. Let us first assume that X 9 V ~ is smooth. Consider the Jacobi tensor A with initial conditions A(a) = 0 and ,4(a) = id. Since the Jacobi equation has no conjugate points, there is a smooth curve Y 9 V such that X(t) = A(t)Y(t). In particular, Y(b) = 0 . Then we obtain I(X, X) -- (X, X)l~i~ -
fh
(~( + R(t)X, X)dt
1'(~4"Y + R(t)AY, AY)dt ---
(2A~+
AY A Y ) d t
2(A );', A Y ) + ( ~ ' , A / A Y ) d t
--2
(~./irar)dt +
(~, (arar))dt.
Since L e m m a 2.2 implies
O _ A / f~ _ f~r A, we obtain
( A r A y ) -- f~l+A y + A r f~ Y + A/+A ) / _ 2 f~/A y + AT+A)/. Hence,
I(X, X) --
(f/,ArAf/)dt.
In the general case, we deduce the same formula by integrating piecewise. This shows I(X, X) ~> 0 for X 9 V (). Furthermore, I(X, X ) - - 0 implies that Y is constant. Since Y(b) = 0, we deduce that 0 = Y(t) = X(t) for all t 9 [a,b]. [7 REMARK. In particular, this l e m m a shows for Jacobi equations without conjugate points on [a, b] that there are no conjugate points inside the interval [a, b] as well. To see this, one considers for a ~ t l < t2 ~< b the piecewise differentiable curve X which is 0 on [a, t i]
Hyperbolic" dynamics and Riemannian geometry
465
and [t2, b] and coincides with a Jacobi field J on [tl, t2] with J (tl) -- J ( t 2 ) - - 0. Then it follows from the formula (2.2) that I(X, X) -- 0 and the Index L e m m a 2.9 implies that X vanishes identically. The next corollary shows that for Jacobi equations without conjugate points, Jacobi fields are not only critical points but minimizers of the action I(X, X) on the space of piecewise differentiable curves with fixed end points. COROLLARY 2.10 (Minimizing property of Jacobi fields). A s s u m e that the Jacobi equation J ( t ) + R ( t ) J ( t ) = 0 has no conjugate points on [a,b]. Let J be a Jacobi field and X ~ V be a piecewise differentiable field with X (a) = J (a) and X (b) -- J (b). Then ~
l(J, J ) <~ I(X, X), where the inequality is strict unless J ~ X.
PROOF. From the Index Lemma 2.9 and the formula (2.2), we deduce 0 ~< l ( J - X, J - X) -- I(J, J ) - 2I(J, X) + I(X, X) -- - l ( J , J ) + I(X, X). 71
This inequality is strict unless J -- X.
In the sequel, we like to compare Jacobi fields associated to a pair of different curvature operators. For that we will need the following partial ordering on S y m ( E ) , where for A, B 6 S y m ( E ) we define A ~< B iff ( A x , x ) <~ ( B x , x ) for all x E E. In the standard Rauch comparison estimates, one of the occurring curvature operators is a constant multiple of the identity. Therefore, the following notation is useful in Rauch's comparison estimates. For k ~ It~ denote by sk(t) 6 C ~ ( I ~ ) the solution of the differential equation ?(t) + k s ( t ) - 0
with initial conditions s(0) - 0 and .~'(0) -- 1, i.e.,
sk(t) --
/
~ - k sinh v/-L-kt
for k < 0,
t ~i- sin x/kt
fork = 0 , for k > 0.
PROPOSITION 2.11 (Rauch). Let R'It~ --+ S y m ( E ) be a C ~ - m a p and J a Jacobi field with J (0) -- 0 and [[J (0)[[ = I. (a) A s s u m e that R ( t ) <<,kl id a n d r <~ 7r/v/kl i f k l > O. Then [[J(t)l[/skl(t) is monotone increasing on [0, r] and b o u n d e d below by !. I f f o r some to > O, we have [[J(t0)[[- skt (t()), then J (t) - sk, ( t ) . J (0) f o r all t 9 [0, t0].
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G. Knieper
468
by applying the above estimate to the solution C ( t ) of the Jacobi equation ~149
C ( t ) 4- R ( r - t ) C ( t ) - - 0
with C(t) -- B ( r - t). Applying the Index L e m m a 2.9 to the curve J(s)-
I A(s)A-I(t)x' I B(s)B-l(t)x,
O<~s<<.t, t<~s<~r,
we obtain 0 < Ii0,,.l(J, J) --(,4(t)A -I (t)x, x ) - (/~(t)B -I (t)x, x) and, hence, the lower bound.
D
LEMMA 2.13. For T >~ 1 and [4 > 0 let R" I~ ~ S y m ( E ) be a curvature operator without conjugate points on [ - 1 , T + 1] and -/4 2 id ~< R(t). Then there exists a constant p > 0 depending only on [3 such that f o r all Jacobi fields J ' [ - 1 , T + 1] --~ E with initial conditions J ( O ) - - 0 and I l J ( O ) l l - 1
IIJ (r)ll >1 p f o r all r e [ 1, TI. PROOF. Fix r e [ 1, T] and consider the piecewise differentiable vector field X ' [ - 1 , 1] ~ E with
X(t) -
0, J(t), (l-(t-r))J(r),
r 4-
- 1 ~
Then O
)-
f
r+l
(R(t)X(t),X(t))dt.
According to Corollary 2.12, we obtain for the Jacobi tensor A with initial conditions A (0) - - 0 and A (0) -- id, the estimate ][/~(r)[[ <~ ][,4(r)A-I(r)l[]lA(r)[] ~ fl[[A(r)[[cothfi. Consequently, IlJ(r)ll ~/~'llJ(r)ll, where fl'--/4 coth ft. This yields for the index form I - - Ii_l,r§ I 0 < I(X, X) ~ IIJ(r)l12(3 ' -4- 1) 4- 3 2
f
r+l
(X(t), X(t))dt
I..a
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G. Knieper
470
LEMMA 2.14. Let R" I~ --+ S y m ( E ) be a curvature operator without conjugate points on [ - 1, oo) and Sr the Jacobi tensor with Sr (0) = id and Sr (r) = O. Then the limit
S(t)-- lim Sr(t) r---->oo
exists and S(t) is nonsingular f o r all t >~O.
PROOF. Since I2(S~,, Sr) = 0, we have that Sr(0) e Sym(E). Furthermore, r ~ Sr(0) is strictly monotone increasing, i.e., (S,. (0)x, x) < (Se (0)x, x) for all x e V, x -r 0 and 0 <~ r < g. Namely, if
X(t)=
Sr(t)x 0
for 0 ~< t ~< r, forr~
and J (t) = Se (t)x, we obtain, using the index form, Ii(}"el and the minimizing properties of Jacobi fields (2.10) -IIo.tI(X, X) --(S,. (O)x, x) < -Ii{},,~l(J, J) --(Se (O)x, x). On the other hand, S,. (0) is bounded above by S-I (0). To prove this, consider
J ( t ) --
S_l(t)x S,.(t)x
for -- 1 ~< t ~< 0, for0 ~< t <~ r.
Using the Index Lemma, we get 0 < ll_,.,.l(J, J) --(S_, (0)x, x ) - (S,.(())x, x). Hence, the limit lim,.~ ~ S,. (0) --" S(0) exists which implies the assertion by the continuous dependence on initial conditions. S(t) is nonsingular for t >/0, since otherwise there are r > 0 and x e E with S ( r ) x = 0. Because there are no conjugate points we would have S,.(t)x -- S ( t ) x for all t ~> 0. This would contradict the fact that (S,.(0)x, x) is strictly monotone increasing. I--1 DEFINITION 2.15. If R" ]t~ ~ S y m ( E ) has no conjugate points we call S(t) = lim S,.(t)
the stable Jacobi tensor, and U(t) =
lim
S_,.(t)
F ----~ Or
the unstable Jacobi tensor.
i
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G. Knieper
472
for all t ~> O. Since lim r---~ ~
sinh c~(r + s) = e us sinh c~r
for a l l s 6 I ~ ,
the inequalities in (2.4) follow. Since S ( - t ) is the unstable solution of , 4 ( t ) + R ( - t ) A (t) = 0, we obtain (2.5) from (2.4). If - f 2 i d ~< R(t) Rauch's comparison estimate implies that I I A ( s ) x l l / s i n h ~ s is monotone decreasing, and we obtain for 0 ~< t ~< r and x 6 E \ {0} IIA(r - t)xll /> sinh/3(r - t)
II A (r)x II
sinh/3r
and IlA(r + t)xll ~< sinh/3(r + t)
IIA (r)x II
sinh f r
Hence, as in the first case, we obtain the estimates in (2.6) and (2.7).
l-q
LEMMA 2.17. Assume that -/4 2 id ~< R(t) and that R has no conjugate points. Then IIS(t)" S-I (t)ll ~ -/~ coth~r. Similarly, we obtain for U,.(t) "= S_,.(t) that /),. (0) <~/~ coth fir. As in the proof of Lemma 2.14 one shows using the Index form I1_,. r I that
o<
x ) - (s
Hence -/-Jcoth/Jr ~< S,.(0) ~< /),-(0) ~< /3coth/4r and, therefore, -/3 ~< S(0) ~< /)(0) ~< /~. F--I Consider the symplectic maps ~Pt,()" E x E ---> E x E with ~t(z) = (J:.(t), J : ( t ) ) , where J is the Jacobi field with (Jr. (0), Jr (0)) -- z. If the Jacobi equation has no conjugate points one defines the stable and unstable spaces E" and E" by
E'={(x,S(0)x)lxeE} and E"--{(x,U(0)x)lx~E}.
473
Hyperbolic dynamics and Riemannian geometry
The following proposition shows the hyperbolicity of the Jacobi equation in the case of a negative definite curvature operator. In particular, this proposition implies that the geodesic flow on a manifold of negative curvature is an Anosov flow (see Section 3). PROPOSITION 2.18. Assume that there are constants 0 < ot < fl such that _f12 id ~< R(t) ~< --0~2 id.
Then E s ~D E u = E x E and there are constants 0 < B <~ A such that ne-t~'llzll ~ II~,(z)ll ~ A e-'*rllzll
for z 9 E", and B e c'' Ilzll ~< II~P,(z)ll ~< A e t~r Ilzll
for z 9 E", where Ilzll- v/llxll e + Ilyll 2 for z -- (x, y) 9 E x E. PROOF. If Z = (x, S(0)x) 6 E ~, we obtain (J:(t), J: (t)) -- (S(t)x, S(t)x) and
]l(J:(t)' J-(t))ll 2 - IIJ:(t)llr "
-+- ( llJ-(t)ll ii~(t)ll )
-IIS(t)xll
v/ (,, 1+
IIS(t).,cll
Since by Lemma 2.17 IIS(t)xll
IIS(t)xll we obtain from Lemma 2.16 e -t~' Ilxll <~ IIJ:(t)ll
% II(J:('), J:('))II <
ll:llV/ +
Since I1=11= v/llxll2-t - IIS(0)xll 2 ~ Ilxllv/1 +/./2 the first estimate follows. The second estimate can be derived in the same way. gl
1.3. Geodesic Anosov flows DEFINITION 3.1. Let ( M , g ) be a (n + l)-dimensional complete Riemannian manifold and II" II the norm on T S M induced by the Sasaki metric. The geodesic flow 49r:SM -~ SM is called Anosov flow if there exist constants k, C > 0 and a splitting
T~,SM = E' (v) 9 E" (v) 9 E~(v)
G. Knieper
474 such that E4~(v) = span{XG(v)} and
IIDr (v)~ II c
e -k' I1~ II
for all ~ 9 E ' (v), t ~> 0, as well as
IID r
(v)~" II ~
c e -k' ][~ 1[
for all ~ 9 E" (v), t ~> 0. REMARK. (a) If M is compact this definition does not depend on the choice of the norm. (b) The definition and the fact that Dgt acts isometrically on the bundle E ~ imply that ES(v) and E"(v) are subspaces of (ECb(v)) -L = N(v). The flow invariance of the symplectic structure co and the inequality Iw(~, r#)[ ~< I1~11" Ilqll imply that E" and E" are isotropic subbundles of N and thus, their dimensions are not bigger than n. On the other hand, E" (v) ~ E" (v) = N(v), and, therefore, the subbundles are Lagrangian. Furthermore, the definition implies that they are invariant and continuous. (c) The definition also implies that I ek t
IIDqb-'(v)~ II ?
li il
for t >~ 0 and ~ 9 E' (v), and [IDdp1(v)~ll
?I
ek t
ii li
for t ) 0 and ~ 9 E"(v). PROPOSITION 3.2. Let ( M, g) be a complete Riemannian manifold with -[42 <~ K <~ -or 2 for constants [3 ~ ot > O. Then its geodesic flow is Anosov. PROOF. Fix an orbit 4t (v) and consider the trivialization Pt :R" x R" ---> N(ck~(v)) of the normal bundle N along 4~~(v), associated to a parallel orthonormal frame Ei (t) . . . . . E , ( t ) orthogonal to 4/(v) (see Section 1 and, in particular, Lemma 1.5). Then Dck~(v)= p~p~p-~i, where lp~ (z) = (Jr.(t), j:(t)) and Jr is the solution of the Jacobi equation f - ( t ) + R(t)Jr.(t) with (J:.(0), Jr.(0)) -- z and Rii(t) -- (R(Ei(t) ~v)dptv, Ei(t)). The curvature assumption implies -/4" id ~< R(t) ~ - o r - i d and from 2.18 we obtain a decomposition of IR" x IR'z into subspaces E" and E" such that B e -t~' Ilzll ~ II~,(z)ll ~ A e -u' Ilzll for z E E", and B e~
~ II~,(z)ll ~ A e~'llzll
Hyperbolic dynamics and Riemannian geometry
475
for z e E". Since Pt is an isometry, the proposition follows for E" (v) -- pt]-I E" and EU (v) = p(-~l E u. 0 The following theorem is due to Klingenberg [60] and shows that only a restricted class of compact manifolds can carry geodesic Anosov flows. THEOREM 3.3 (Klingenberg). Let (M, g) be a compact Riemannian manifold such that the geodesic flow is Anosov. Then there are no conjugate points. REMARK. Together with the Theorem of Hadamard and Cartan (see Section 2, Theorem 1.1) it follows that the universal covering of compact manifolds carrying an Anosov geodesic flow has to be diffeomorphic to IR". It is an amazing result due to Marl6 that the conclusion of Theorem 3.3 holds under a much more general assumption [70]. THEOREM 3.4 (Marl6). Let (M, g) be a complete Riemannian manifold o f finite volume. Assume that the bundle N -- ( ECb)• normal to the geodesic flow qbI admits a continuous Lagrangian subbundle invariant under the geodesic flow. Then (M, g) has no conjugate points. IDEA OF THE PROOF. The existence of a Lagrangian bundle can be used to associate to any closed continuous curve c: I ~ S M an index which counts the crossing of the Lagrangian bundle with the vertical bundle V,, = {(0, w) l w _k v} (Maslov index) [26]. The index does not change under homotopic deformations of c. Using this fact, Marl6 proved that the intersection of the Lagrangian bundle with the vertical bundle is trivial, i.e., the bundle is a Lagrangian graph. Hence, in view of Lemma 2.7 there are no conjugate points (see [70] for more details). [-1
In the same paper, Marl6 claims a generalization of Klingenberg's theorem to noncompact manifolds with geodesic Anosov flows provided the sectional curvature is bounded from below. However, Burns noticed a mistake in Mafi6's arguments [17]. Since certain statements remain valid and might be useful for future research in this area, we will briefly sketch his ideas. Denote by C ( S M ) = {v e S M I c,, :[0, oo) -+ M has no conjugate points} the set of vectors in the unit tangent bundle corresponding to geodesics with no conjugate points. It is not difficult to prove that for all Riemannian manifolds C ( S M ) is a closed subset of S M . Moreover, this set is nonempty if M is not compact. Under the assumptions that the geodesic is Anosov, Marl6 claimed that this set is open as well. The proof is based on Proposition II.2 of his paper which, unfortunately, is not correct as stated there. However, the following weaker version is true. It is a consequence of the Jacobi field estimate given in Lemma 2.13. LEMMA 3.5. Let (M, g) be a Riemannian manifold with lower sectional curvature bound. If the geodesic flow is Anosov there exists a constant cr with the following property. If E" (v) A V (v) r {0} then the geodesic cv has conjugate points on the interval [ - 1, o'].
G. Knieper
476
PROOF. Suppose the geodesic c v ' [ - 1 , T + 1] -+ M, T ~> 1, has no conjugate points. Consider 0 # ~ = (0, ~2) E E s (v) fq V (v) such that 11~2[I = 1. If J is the Jacobi field with J (0) = 0, J ' ( 0 ) = ~2 then, according to L e m m a 2.13 and the definition of E s, p ~< IlJ(t)ll ~< V/IIJ(')II 2 + IIJ'(t)ll 2 - - I I D ~ ' ( ~
II ~
Ce-k'
for all t e [ 1, T]. Therefore, T ~< (log C - log p ) / k -- ?' and the geodesic cv" [ - 1, o-] --+ M has conjugate points provided o" > Y + 1. 17 REMARK. Marie's mistake consists in the claim that, under the assumption of L e m m a 3.5, the geodesic c,, should even have conjugate points on [0, cr ]. From that it is easy to deduce that a geodesic c~, : [0, cxz) --> IR has no conjugate points provided it has no conjugate points on [0, or]. This would have shown that the set C ( S M ) is open. OPEN PROBLEMS.
(a) Is it true that also noncompact manifolds with geodesic Anosov flow do not have conjugate points? (b) It follows from the discussion in [60] that the universal cover of a Riemannian manifold with geodesic Anosov flow is hyperbolic in the sense of Gromov [42]. Is it true that each compact manifold with a geodesic Anosov flow carries a metric of negative curvature'?
2. Geometry of manifolds of nonpositive curvature 2.1. Hadamard man(folds We begin with the following theorem which is foundational in the theory of Hadamard manifolds (see, e.g., [21,53,39,61,80]). THEOREM 1.1 (Hadamard-Cartan). Let ( M , g ) be a complete n-dimensional Riemannian man~fold with no conjugate points. Then, for all p E M, the exponential map expp " TI, M -+ M is a covering map. In particular, the universal covering of M is diffeomorphic to IR". REMARK.
(a) The topology of those manifolds is to a large extend determined by the fundamental group since the contractibility of the universal cover implies that the higher homotopy groups are vanishing, i.e., n'k (M) -- 0 for k ~> 2. (b) If M i, M2 are compact manifolds with nonpositive curvature such that 7rl (M l) and n'l (M2) are isomorphic, then, by a result of Farell and Jones, Mi and M2 are homeomorphic. However, in general they are not diffeomorphic [29,30]. DEFINITION 1.2. A Hadamard manifold is a simply connected complete Riemannian manifold of nonpositive curvature.
Hyperbolic dynamics and Riemannian geometry
477
REMARK. According to Theorem 1.1, Hadamard manifolds are diffeomorphic to R". Furthermore, there is a unique geodesic connecting any pair of points. The most important single tool in the study of Hadamard manifolds are convexity properties of the distance function. A function f : X ~ IR is called convex if for all geodesics c : R ~ X the composition f o c :JR ~ I~ is convex in the usual sense. A set B C X is called convex if for any pair of points p, q 6 B the connecting geodesic is contained in B. In particular, if f : X --+ IR is convex the set {x E X I f ( x ) ~< oe} is convex for all c~ ~ ]R. The following are examples of convex functions. Let d : X x X ~ IR be the distance function on a Hadamard manifold X induced by the Riemannian metric. (a) If cl, c2 : N ~ X are geodesics then t ~ d(cl (t), c2(t)) is convex. (b) If B C X is convex then the function x ~ d ( x , B) is convex. (c) Busemann functions are convex (see Section 2.3). PROOF. The statements in (b) and (c) are consequences of (a), whereas (a) follows from the second variation formula of arc length. I-7 Denote by
,.)
(b) f~ ~> f i + f;_ -- 2glf2cosc~3. (c) g3 ~< g2 cosot'l + gl cosc~2. In each case equality holds if and only if the sides o f the triangle span a totally geodesic flat triangle. COROLLARY 1.4. Let Pt, P2, P3 and p4 be distinct points and consider the quadrilateral with angles Oil --
As a consequence one obtains the so-called flat strip theorem. THEOREM 1.5. Let ct, c2" IR ~ X be two geometrically distinct geodesics. If d(cl (t), c2(t)) is bounded by a constant independent o f t the geodesics span a flat totally geodesic embedded strip. PROOF. Since t --> d(cl (t), c2(t)) is convex and bounded it has to be constant. Then one obtains from the second variation of arc length that for all tl < t2 the points cl (tl), c2(tl ), cl (t2), c2(t2) span a quadrilateral of angle sum 2rr. Using 1.4, one obtains the result. 1--1
G. Knieper
478
The asymptotic geometry of Hadamard manifolds is intimately related to the long time behavior of the geodesic flow and plays a central role in rigidity theory. We begin by introducing the following fundamental notions. DEFINITION 1.6. Let ( X , d ) be a metric space and A C X. For r ~> 0 the r-tubular neighborhood of A is the set
Tr(A) := {q E X I d(q, A) <~ r}. If A, B C X are subsets of X,
dH(A, B ) = inf{r I A C T,.B, B C TrA} denotes the Hausdorff distance of A and B. REMARK. It is easy to see that dH fulfills the triangle inequality. DEFINITION 1.7. Let X be a Hadamard manifold. We call two geodesics cl : R X, c2 : R ~ X asymptotic if cl.(,'.(R+).c2(R+))
< ~.
DEFINITION 1.8. The set of equivalence classes of asymptotic geodesics is called the sphere at infinity and denoted by Xgc,,d(cx~), or simply X ( ~ ) , if there is no confusion of the meaning. For p e X consider the map .ft, :SI, X -+ X(cx~) such that v --+ [c,,], where c,, is the geodesic with b,,(0) = v and, where [c,,] denotes the equivalence class of c,, :II~+ ~ X in X(cx~). This map is a bijection. The topology on X ( ~ ) , defined such that fl, becomes a homeomorphism, is called the sphere topology. Since fi/ o ] ) , l ' S i , X ~ SqX is a homeomorphism this topology is independent of p. The topologies on X(cxz) and X have a natural extension to X := X t2 X (c~) and are characterized by the fact that the map
~: B~ ( p ) - {v ~ T~,M I Ilvll ~ 1} ~ X with 99(v)--
{ (' t expp
fp(v),
I-Ilvtlv ,
[Ivii < 1,
Ilvll- 1,
is a homeomorphism. This topology has been introduced by Eberlein and O'Neill [28] and is called the cone topology. In particular, X is homeomorphic to a closed ball in ~ " . The relative topology on X(oo), respectively, on X coincides with the sphere topology, respectively, the topology on the manifold X.
Hyperbolic dynamics and Riemannian geometry
479
2.2. Rigidity o f the sphere at infinity in negative curvature For Hadamard manifolds of strictly negative curvature the sphere at infinity has amazing rigidity properties first discovered by Morse and further investigated by Gromov culminating in the work of Gromov hyperbolic metric spaces [42]. In these spaces it is possible to describe X(oo) using quasi-geodesics which are objects much more diffuse than geodesics. Their advantage, however, is that they are much more robust than geodesics. For example, a quasi-geodesic for a given metric will still be a quasi-geodesic for all biLipschitz-equivalent metrics. Let us first recall the notion of a quasi-isometry. DEFINITION 2.1. Let ( X i , d l ) and (X2, d2) be two metric spaces. A map F" (XI, dl) -+ (X2, d2) is called a quasi-isometric embedding if and only if there exist constants A > 1, c~ > 0 such that 1
--dl (x, y) - c~ ~< d2(Fx, F y ) <~ Adl (x, y) + c~. A A quasi-isometric embedding F is called a quasi-isometry if it has a quasi-isometric inverse G" X2 ~ X! such that
d,(GoF(x),x)<~p
and
d2(FoG(y),y)~
for a constant p > 0 and all x E X i and y 6 X2. In this case, X I and X2 are called quasiisometric. A quasi-isometric embedding q9" (R, l I) ~ (X, d) is called a quasi-geodesic. REMARK. If we want to specify the constants we call such maps (A, c~)-quasi-isometric embeddings ( ( A, c~)-quasi-isometries, ( A, c~)-quasi-geodesics). Let X be a Hadamard manifold. We extend the equivalence relation of being asymptotic to the much larger class of quasi-geodesics. Denote by the equivalence classes of quasi-geodesics, and by Xgc,,d(oo) the equivalence geodesics. Denote by [ ]gc,,d, [ ]quasi the corresponding equivalence classes. The following theorem implies that we do not loose information if we geodesics to quasi-geodesics.
geodesics Xqua~i(oo) classes of pass from
THEOREM 2.2. Let (X, g) be a Hadamard manifold such that K e <, - k 2 < O. Then the inclusion Q " Xgc,,d(oo) ~ Xquasi(oo) with [C]geod ~
[C]quasi is bijective.
G. Knieper
480
It is obvious that Q is a well defined injective map. The nontrivial part is to show that to each quasi-geodesic ~0 there is a geodesic representative c, i.e., dH(~(R+), c(R+)) < oo. We will prove a somewhat stronger result which is a modern version of a lemma by Morse (Morse lemma) [74]. THEOREM 2.3 (Morse). Let X be a H a d a m a r d manifold o f negative sectional curvature K <. - k 2 < O. Let qg:~, ~ (X, dg) be a (A, ot)-quasi-geodesic. Then there exists, up to parametrization, exactly one geodesic c such that dH(c(It~), qg(II~)) <~ ro, where ro depends only on A, ot and k.
The proof will be a consequence of the following considerations. Let c : R --+ X be a geodesic and P,. : X --+ c ( R ) be the orthogonal projection, i.e., P,. is the identity on c('~) and for p r c(R) the projection P,,(p) is the unique element of c(/R) such that the geodesic segment connecting p and P,.(p) is perpendicular to c(R). By the convexity of geodesic balls, P,. is well-defined and has the following properties. LEMMA 2.4. Let X be a Hadamard manifold o f negative sectional curvature K <. - k 2 < O. Then we have: (a) For all p e X d(p,c(R))=d(p,P,.(p)).
(b) P,. is differentiable and IIDP,.(p)vll <~
c o s h k d ( p , P,.(p))
Ilvll
f o r all p e X and v e T I, X.
(c) For p > 0 there exists y = y ( p , k) > 0 such that f o r all p e X we have diam P,.(B(p, p)) <~ y e -k'I(p'pIpl). PROOF. Statement (a) follows from the convexity of geodesic spheres and (b) follows from the upper curvature bound together with a generalization of Rauch's comparison theorem (see 2.11). Statement (c) is a consequence of (b). D LEMMA 2.5. Let tp : • --+ X b e a n (A, ~)-quasi-geodesic and ot' = u + there exists xl . . . . . x,, e qg([a, b]) such that d ( x i , x i + l ) ~ 2Or' and qg([a, b]) C 0
1. Then, f o r a < b,
B(xi, 2ct'),
i=1
where n = [A(b - a)/ot'] and [r] denotes f o r r e R the largest integer smaller than r.
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P R O O F . L e t a < tl < . . . < tn <~ b be the a ' / A equidistant subdivision of [ a , b]. Since ~o is a (A, c~)-quasi-geodesic, we have for all s E [a, b]
d (qp(s), (p(ti)) ~ or' q-ot < 2~' (.gt
"' where ti = i -X + a is provided we choose 1 <. i <~ [ A ( b - a ) / ~ ' ] such that Is - ti[ <~ -X, an element of the subdivision introduced above. Consequently, xi = cp(ti) are the required elements of ~0([a, b]). I--] LEMMA 2.6. Let ~o : lR --+ X be an (A, c~)-quasi-geodesic a n d c : IR --+ X be a geodesic. Then there exists a constant ro(A, or, k) with the f o l l o w i n g property: For all a < b a n d r >~ ro(A, ~, k) such that cp([a, b]) C X \ TrC(]R) and
d(~o(a),c(IR)) <. r
+ oe
and d(~o(b),c(IR)) <<.r
+
~,
we have: diam(P,.~o(la, bl)) ~< 1.
PROOF. Lemma 2.5 implies the existence of xl . . . . . . ~c,, E ~o([a, b]) such that
cp([a, b]) C 0
B(xi, 2c~'),
i=1
where oei --oe + 1 and n <~ ~A( b - a). Using Lemma 2.4 and the lower bound in the definition of an (A, c~)-quasi-geodesic, we obtain
i=1
A2 A Ib - a l y e -k' <<"7 (d(~o(a) ' Io(b)) + o~)y e - k ' <~ ny e -k'' <~ -~7 The assumption, together with the triangle inequality, implies d(~o(a), ~o(b)) <~ 2r + 2or + I.
Consequently, there exists a constant r0(A, a~, k) such that A2 A2 g ~< -~-7(2r + 3a~)V e - k ' + otye?' e - k ' <~ 1/2 + e/2 for all r ~> r()(A, or, k).
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Hyperbolic dynamics and Riemannian geometry
483
PROOF OF THEOREM 2.3. Let qg" R -+ X be an (A, ct)-quasi-geodesic and let t, ~ I~ be a sequence such that t,, -+ cx~ as n --> cx~. L e m m a 2.7 implies that for any sequence of geodesics c, "]K --> X with c , ( a , ) -- 99(-t,,) and c ( b , ) -- r the estimate dH(qg([--t,,, t,,]), c([a,,, b,,])) ~< rl
holds. Choosing a parametrization of the geodesics such that
c,, ( 0 ) - Pc,
(~0(0))EBr, (~0(0)),
we conclude that a, -+ - e ~ and b, ---->+oo. Furthermore, there exists a convergent subsequence b,,i (0) of g',, (0) converging to some vector v E S X. Then dn(c,,(]K), r
where c,, " R ~
~< rl,
X is the geodesic with b,,(O) -- v.
I-1
As a conclusion one obtains" THEOREM 2.8. l_z,t (X, gl) be a H a d a m a r d manifold and (Y, g2) a H a d a m a r d man~fold with curvature K e2 <~ - k 2 A s s u m e that F" X ~ Y is a quasi-isometric embedding. Then F induces a continuous injective map F ~ 9 X ( o c ) ~ Y(cx~), defined by F..~ ([,']gc,,d) -- Q - ' ([ F o C]qua~i). COROLLARY 2.9. Let (X, gl) and (Y, g2) be two H a d a m a r d man~folds such that K~ i <~ - k 2 < O f o r i E {1,2}. If X and Y are quasi-isometric, X ( o ~ ) and Y(o~:~) are homeomorphic. Note, that the universal coverings of compact homotopy equivalent Riemannian manifolds are quasi-isometric. This even holds in the case of geodesic length spaces, i.e., complete inner metric spaces. More precisely, one has: PROPOSITION 2.10. Let ( M i , d t ) a n d (M2, d2) be two c o m p a c t homotopy equivalent geodesic length spaces and assume that their universal coverings X t a n d X2 exist. Then, with re.wect to the lifted metrics, X t and X2 are quasi-isometric. COROLLARY 2.1 1. Let ( M t , g l ) a n d (M2, g2) be c o m p a c t Riemannian manifolds o f negative curvature with isomorphic f u n d a m e n t a l groups. Then the ideal boundaries X I (r a n d X2(oo) o f their universal coverings are homeomorphic. PROOF. Note that compact K(zr, 1) manifolds Ml, M2 with isomorphic fundamental groups are homotopy equivalent. I-1
G. Knieper
484
A further application, due to Gromov [40], shows that geodesic flows on compact manifolds of negative curvature are orbit equivalent, provided their fundamental groups are isomorphic. One can consider this as a global version of structural stability (see [2]). THEOREM 2.12. Let (Mi, gl) and (M2, g2) be compact Riemannian manifolds o f negative curvature with isomorphic f u n d a m e n t a l groups. Then the geodesic flows dpl,I " S M I --~ SMI and q~tq,2"SM2 --+ SM2 are conjugate (not time preserving). More precisely, there exists a homeomorphism F : SM! ---> SM2 such that
For where T : I~
x
",~2
o F(v),
SMI --> IR is the corresponding time change.
PROOF. L e t F : X! ----> X2 be a lift of a homotopy equivalence and F ~ : Xi (00) ~ X2(oo) be the homeomorphism induced by F. Consider the map G(): S X I ---> SX2 defined by
Go(v) := PF~(,, ).F.,.(,,t )F(rrv), where v + = c,,(+oo) 6 Xi (oe) and for ~, tt 6 X2(oo), we denote by P~.t, the orthogonal projection onto the oriented geodesic determined by ~ and l~. More precisely, P~./~ (x) = b~./~(t) if P , ( x ) = c~.lL(t), where c is a geodesic with c ( - o o ) = ~ and c(+cx~) = tt and P,. is the ordinary orthogonal projection on the geodesic (' introduced above. This map is equivariant and surjective but in general not injective. To obtain a homeomorphism consider the map s : IR x SXI --* I~ defined by
G,,(~'~,,, ~1 -
_d~,<1.,')G()(v) ~
which is an additive cocycle, i.e.,
.,.<~ + t, ~)- .,.(,, ~',,, ~)+ .,.~t, ~). Choose r > 0 such that s(r, v) > 0 for all v 6 SXI and consider the average s(t, v) dt.
r ( v ) -- l"
Define Gr : S X I ---> SX2 by GT(v)-
r
G()(v).
This map is equivariant and, therefore, descends to a map S MI ---> S M2 preserving the geodesic foliations. Furthermore, Gr is injective since
,-/4,/,~~,)c,,(,pl,., v) -,~.~r (4,f~,I v)+.~'(/,v) c,,~,,)
c, (~'I,.,v) -,~.~2 '
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and
m(1)
=
r(dpl~,,v ) + s(l, v) '
- - -1
fo r
s(t
+ l, v)
dt
"t"
is strictly monotone increasing. The monotonicity follows from m , (1) -- -l f t r
s , (t + 1, v l d t = -1( s ( r + I, v) - s(l, v)) -- -1s (r , qS/,o)>O.
l"
1"
Z"
E]
2.3. B u s e m a n n f u n c t i o n s and the horospherical foliations Busemann functions play an important role in the asymptotic geometry and the dynamics of geodesic flows on Hadamard manifolds. They also can be used for an alternative compactification of Hadamard manifolds (see [27] for more details). LEMMA 3.1. Let X be a H a d a m a r d manifold and c : IR --+ X be a geodesic" representing c X (oo) with c(O) = y. Then the limit
b, (x, ~) := lim d ( x , ~'(t)) - t I ---~ %
exists and de[ines a conve~ C2-function. Furthermore, its gntdient has norm I.
PROOF. See I481.
[--]
REMARK.
(a) We remark that the levels of x ~ d(x, c(t)) - t are spheres about c(t). The zero level passes through y. (b) Note, that on any Riemannian manifold ( M , g) the PDE II grad f I1., = I is the Hamilton-Jacobi equation associated to the Hamiltonian G ( v ) ~g(v, v). The integral curves of this gradient field are geodesics parametrized by arc-length. We also remark that CI-functions f with II grad f I1,~,= 1 are of class C I'l , i.e., they have a Lipschitz derivative [31]. DEFINITION 3.2. Let X be a Hadamard manifold. For y ~ X and ~ ~ X ( ~ ) x ~ b,.(x, ~) is called the Busemann function associated to y and ~. The levels of this function are called the horospheres based at ~ 6 X ( ~ ) . We denote by Hor(x.t) the horosphere based at ~ passing through x 6 X. If v ~ S M we denote by Hor,, the horosphere based at c~,(oo) 6 X (c~) passing through c,,(0).
486
G. Knieper
REMARK. Since by I (x, ~) - by 2 (x, ~) = by, (Y2, ~) the levels only depend on ~ and not on y. In particular, the gradient grad b v ( x , ~) is independent of y and we denote it by B ( x , ~). For each x, by(x, ~) is the signed distance of x from the horosphere through y. It is decreasing if we move towards ~ and increasing if we move away from ~. From a dynamical point of view the stable and unstable foliations in S X are of particular importance. They are called the horospherical foliations. By W 's and W TM we denote the weak stable and weak unstable foliation, i.e., the foliation of S X whose leaves through v E S X are given by
w ' ' (v) - {w 9 s x I c , , ( + ~ ) = c,,,(+oo) } and w('li ( ~)) -
[ ~l.) E S X l c u ( - (~) ) = c|/,(-(x~))].
Note, if se = c,,(+c~), rl = c , , ( - c ~ ) and y E X, we have W"'(v)-
{ - g r a d b , . ( x , ~ ) I x E X]
and
W'"'(v)- {gradb,,(x, ,1)[x E X} and, therefore, the restrictions 7r:W'"'(v)---> X, zr: W'""(v)---> X of the canonical projection zr : S X -+ X are diffeomorphisms. By W" and W" we denote the strong stable and strong unstable foliation. The leaves through v E SX of those foliations are given by
W"(v)- [-gradb,,(x, ~)[x E Hor,,}, W" (v) -- {gradb,,(x, '1) Ix E Hor_,, },
where ~ = c,,(+oo), r / = c , , ( - ~ )
= c_,,(+o~:~).
The tangent spaces of W " ( v ) and W ' ( v ) can be described as follows. Let S,,(t) and U,,(t) denote the stable and unstable Jacobi tensor along c,, (see Section 1.2, Definition 2.15). Define S ( v ) " - - SI,(O) and U ( v ) " = U,',(0). Note, that U ( v ) and S ( v ) are the second fundamental form for the unstable and stable horosphere. More precisely, U ( v ) : v • ~ v • (v • = {w E T~r~,,)X [ (w, v) = 0 } ) is the symmetric endomorphism with U(v)w--
V,,, gradby (zr(v), q),
where 77= c , , ( - c ~ ) . Similarly, S ( v ) :v • ~ S ( v ) w -- -V~,, gradby (~r (v), ~),
v • is the symmetric endomorphism with
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487
where r = c v ( ~ ) . Using the decomposition of T SX into horizontal and vertical spaces (see Section 1.1), one has:
and
2.4. Volume growth and entropy formulas Important invariants which reflect the complexity of a dynamical system on an exponential scale are the topological and the measure-theoretic entropies. The topological entropy is an invariant of the topological dynamics and measures in some sense the maximal complexity of the dynamical system. In particular, complexity on small regions can cause positive topological entropy. On the other hand, measure-theoretic entropy is an invariant of the measurable dynamics reflecting the average complexity of the system relative to an invariant measure. If f : X --~ X is a homeomorphism of a compact metric space X, the topological entropy ht,,p(f) is always bigger or equal than the measure-theoretic entropy h1~(]') for any given invariant Borel probability measure 1~. The variational principle (see [2]) implies that for all f-invariant probability measures .h4 ( f ) ht,,p(f) -- sup{h,,(]') ll~ ~ M ( f)}. Measures for which this supremum is attained are called measures of maximal entropy. In Section 5 we will show that certain geodesic flows on manifolds of nonpositive curvature have a unique measure of maximal entropy. Note, that for flows the entropy is defined as the entropy of the time-I map (see [2] or [57,83]) for a comprehensive introduction to entropy theory. In differential geometry there is an asymptotic quantity closely related to entropy. It measures the exponential volume growth on the universal cover of a compact manifold and is defined as follows. Let (M, g) be a compact Riemannian manifold and X be its universal covering. For p 6 X denote by B(p, r) the geodesic ball of radius with respect to the lifted metric. It is a consequence of the compactness of M (observed by Manning [71 ]) that the limit lim r ---* ."y,.)
logvolB(p,r) r
exists and is independent of p ~ X. We denote this limit by h(g) and call it volume entropy. It measures the exponential volume growth of balls in the universal covering. As has been shown by Manning, this number is related to the topological entropy htop(~) of the geodesic flow cpI:SM --~ SM.
488
G. Knieper
THEOREM 4.1. Let (M, g) be a compact Riemannian manifold. Then htop(~)/> h(g). If (M, g) has nonpositive curvature then htop (05) -- h (g). REMARK. Freire and Marl6 [36] showed that equality even holds if the metric has no conjugate points. In the case of negative curvature there is an interesting formula expressing the volume entropy as an average of certain curvature quantities. The measure /zH involved in this average is locally of the form d # H = d,k" d/z'", where ,k" is the Lebesgue measure on the strong unstable foliation and/z,.s is the Margulis measure on the weak stable foliation (see Section 3.1). This measure is also obtained as a limit of spherical averages (see [63]). The following entropy formulas are derived in [63]. THEOREM 4.2. Let (M, g) be a compact Riemannian manifold of negative curvature, then h2(g) -- f M Ric(v) - scal o rr + s(v) d/ZH, where Ric and scal are the Ricci and scalar curvature of M, and s(v) is the scalar curvature of the horosphere through v ~ S M. For surfaces the formula specializes to h2(g) - f ' M - K o n d/ztt, where K denotes the Gaussian curvature of M. For 3-dimensional manifolds he(g) - ~'M Ric(v) - scal o Jr d/z//. There are also estimates and formulas for the measure-theoretic entropy on manifolds with no conjugate points. Using Ruelle's inequality and Pesin's formula for the measure entropy of smooth invariant measures [78,79], Freire and Marl6 obtained the following estimate for the measure theoretic entropy of the geodesic flow. THEOREM 4.3. Let (M, g) be a compact manifold without conjugate points and/Z be a ckt-invariant probability measure for the geodesic flow. Let U(v) denote the second fundamental form for the unstable horosphere (see Section 2.3). Then ht~ (~b) <~ f ' M trU d/z. Equality holds if # is a smooth measure.
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2.5. Poincard series and Busemann densities Let (X, d) be a metric space and F C Iso(X) be a discrete infinite subgroup of the isometry group of X. For p, q E X and s E R consider the Poincar6 series e-sd(p,Yq)
P(s, p, q) = Z vEF
For k E I~ consider
a k ( p , q ) = c a r d { y E F Ik ~ d ( p , yq) < k + 1} and define 1
6(p, q) -- k lim (p, q) ---~cy.J -~ a k LEMMA 5.1. 6 is independent of p and q. Furthermore, P(s, p, q) converges for s > 6 and diverges for s < 6. DEFINITION 5.2. The number 6 = 6 ( F ) is called the critical exponent for/-'. The discrete group F is called of divergence type if the Poincar6 series diverges for s = 6 ( F ) . In case that F is a uniform lattice in a Hadamard manifold X, i.e., X / F is compact, we have the following simple criterion for the divergence. LEMMA 5.3. Let M = X / F
be a compact manifold of nonpositive curvature. Then P(s, p, q) diverges for s = 6( F) ~f and only if
L
~ e -'5' vol S(p, r) dr
diverges for one, and, hence, any point p E X. PROOF. Let .T be a fundamental domain for F . Then the divergence of P(s, p, q) is equivalent to the divergence of
f~ ~
e -''l(p'•
dvol(y) =
e -''lIp'.v) dvol(y) --
e - " ' vol S(p, r) dr.
y E l ~"
I-1 REMARK. If M is compact this implies that 6 ( F ) = h(g). In Section 5 we will see that all cocompact actions of F on nonpositively curved spaces are of divergence type.
G. Knieper
490
Let X be a H a d a m a r d space and F C Iso(X) be a discrete subgroup of the isometry group of X. For x e X denote by A ( F ) the accumulation points of the orbit F x in X. This set is given by Fx A X (oo) and is independent of x e X. It is called the limit set of F . If F is cocompact, i.e., X / F compact, then A ( F ) = X(cx~). Next we introduce a certain family of measures on X (cx~), supported on the limit set of the discrete group F which is important from a dynamical and geometrical point of view. This family of measures is called B u s e m a n n densities or conformal densities. DEFINITION 5.4. Let X be a H a d a m a r d manifold and F C Iso(X) be a discrete subgroup. A family of finite Borel measures {/Zp} p~X on X (cx~) is called an c~-dimensional B u s e m a n n density if (a) supp/zp C A ( F ) for all p 6 X, (b) ~dl~q (se) -- e _Otbl , (q,~ ) for almost all ~ e X (oo), (c) {#p}p~X is F-equivariant, i.e., # •
lZp(A) for all Borel sets A Q X ( o o ) .
There is a standard way to construct such a measure. In the case of Fuchsian groups this is due to Patterson [77]. For general hyperbolic spaces it has been intensively studied by Sullivan [82]. We briefly describe this construction. A s s u m e that the group F C Iso(X) is of divergence type. If not, one introduces a positive m o n o t o n e increasing function f :IR + --+ I1~+ such that for each a > 0, we have that f ( r + a ) / f ( r ) ~ 1 as r -+ oo and that the modified Poincar6 series
,~(s, x,
y)- Z f(d(x, yy))e -'''(''• y~l"
diverges for s - 3 ( F ) . Fix x 6 X and for s > 6 and p 6 X consider the measure
e-Sd(p,yx t s p , x , .~' " - -
P(s,x,x)
where 6~. is the Dirac mass associated to y e X. ~tp..,..,. has the following obvious properties. (1) e - ' a / / ' ' ' t <~ p p, ,.,.,.(X) <~e ''lIp''). In particular, the measures are finite. (2) Fx C supp /t p, ,...,. C Fx. Now choose for a fixed p e X a weak limit limk--. ~ kt p,x,.,k =" ktp. The divergence of the series P(s, x, x) for s = 6 and the discreteness o f / - ' yields that the support o f / z p,.,..,k moves toward the limit set if k ~ cx~. More precisely, one obtains" PROPOSITION 5.5. For each q e X, the weak limit l i m , . k ~ lZq,x,.,.k = : #q exists. The family of measure {ktp} p~x is a 8(F)-dimensional Busemann density. REMARK. (a) A B u s e m a n n density constructed in such a way is also called a Patterson-Sullivan measure.
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491
(b) There is up to a constant only one such finite measure if X / F is compact and X has negative curvature, or more generally, is a nonpositively curved rank 1 manifold (see Section 5.1).
3. Horospherical foliations 3.1. Ergodic theory of the foliations In his thesis Margulis [72] constructed for the stable and unstable foliations of an Anosov flow cpt : V ---+ V a measure which is very useful in geometrical and dynamical applications (Margulis measure). Let W'" denote the weak stable foliation of the Anosov flow. Then there exists a family of measures {tt'~"}x~v on the leaves W"*(x) which changes continuously if we slight it along the strong unstable foliations. Furthermore, it contracts exponentially under the geodesic flow on a fixed leave W'' (x). More precisely, cs
/z x o r
t
=e
-hi
cs
#x,
where h = ht,,p is the topological entropy of the flow Ct. Those properties even apply that these measures {~'~:'}.,~v define an invariant transversal measure for the strong unstable foliation. Margulis obtained this measure using a fixed point argument. Bowen and Marcus [15] showed that there exists only one invariant transversal measure for the strong stable foliation. It relies on the Markov property of the foliation. For Anosov diffeomorphisms there is an easy proof due to Fathi, Laudenbach and Po6naru [32]. Since each weak stable leave W'"' (x) is foliated by strong stable leaves, namely W'' (x) = U,~]~ cPt W" (x) one can disintegrate the measure and obtains dtt.'~)"(r y) -- e -h' dt d/~:'i.(y) for all y E W"(x). Similarly one constructs measures {/~i~-"}.,-~v on the weak unstable foliation W'"' with analogous properties, namely lz cu .v
o
~t
--
eht ll, xcu .
Disintegrating [)'.l 9 "" analogous to tti~." yields measures/z.',! on the strong unstable foliation In the case of compact manifolds M with negative curvature one obtains those measures also using the Patterson-Sullivan measure {/zp}p~x constructed in Section 2. Fix a leave W" (v) of the strong stable foliation and consider the bijection P,, : W" (v) --+ X (c~) \ {se }, where ~ = c,,(oo) and P~,(w) = c,,,(-oo). For u, ~ W' (v) define d/]~,(w) "-- dgJrI,,,)(Pv(u')) = e -h/'~''')(rr''''~) dtt~r(,,)(P,,w). This obviously defines a measure as stated above. Using the transformation properties of the Patterson-Sullivan measure one observes that this defines a measure stated above. By uniqueness d~,'i -- d#i,.
492
G. Knieper
Using those measures, Margulis constructed a 4~t-invariant measure of maximal entropy (see Section 2.4) which is nowadays called Bowen-Margulis measure and is, therefore, denoted by #BM. Locally #rim is of the form d#xII d ~x' ' 9 Bowen gave another construction of a measure of maximal entropy as a weak limit of measures supported on periodic orbits [13]. Furthermore, he showed [ 14] that the measure of maximal entropy is unique for Anosov flows which, in particular, implies that both constructions lead to the measure. It turns out that the existence and uniqueness of the measure of maximal entropy even holds in the case of geodesic flows on certain compact nonpositively curved manifolds with some amount of hyperbolicity [66]. In Section 5 we study the dynamics of such geodesic flows and give a sketch of the proof of the uniqueness. For a general survey on stable and unstable foliation see [1 ].
3.2. Regularity o f the foliation and rigidity E. Hopf [51] was the first who started to investigate the regularity of the horospherical foliation on a compact surface of variable negative curvature. He obtained that the foliation is of class C I . This was sufficient to apply a "Fubini"-type argument, which is nowadays called the "Hopf argument", to prove ergodicity of the geodesic flow with respect to the Liouville measure. Hurder and Katok [52] proved that for surfaces the foliations are even of class C I'~ll''~-~l which, in particular, implies C I''~ for all c~ < 1. We remark that C I'~ is also deducible from the work of Hirsch, Pugh and Shub [50]. In higher dimensions Green [38] proved C I regularity in the case that the curvature is quarter pinched, i.e., - 4 a ~< K ~< - a for a > 0. This has been also obtained by Hirsch and Pugh [49] using the techniques developed in [50]. In 1966 Anosov showed, for general manifolds of negative curvature (or even general Anosov flows (see [4])) the absolute continuity of the foliation with respect to the Liouville measure. This is still enough to apply Hopf's argument for proving ergodicity. Already Anosov realized [4] that there are obstructions for having high regularity (Anosov cocycle). Those obstructions have been thoroughly studied by Hurder and Katok [521 in the case of geodesic flows on compact surfaces (or more generally for contact Anosov flows on 3-manifolds). They showed that C I'l regularity already implies C ~ . On the other hand, Ghys [37] proved that in dimension 3 contact Anosov flows with C ~ foliations are smoothly conjugate to an algebraic flow, in particular, the Liouville entropy and the topological entropy coincide. For surfaces of negative curvature this implies constant curvature by an entropy rigidity result of Katok (see Section 4, Corollary 1.1). Kanai [55] extended Ghys' result to negatively curved manifolds of higher dimensions. His main tool was a certain connection on the unit tangent bundle called the Kanai connection (for the definition see the discussion below). He showed that the geodesic flow of compact negatively curved spaces (M, g) with C ~ Anosov foliation is time preserving conjugate to a geodesic flow on a compact manifold of constant negative curvature, provided - 9 / 4 < K ~< - 1 . Using deeper dynamical arguments Feres and Katok [34] obtained the same result under the optimal pinching condition - 4 < K ~< - 1. Later on, Feres [33] and Benoist, Foulon and Labourie [9] studied the compact negatively curved manifolds without pinching assumptions. The most general result has been obtained by Benoist, Foulon and Labourie.
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THEOREM 2.1. Let M be a compact manifold of negative curvature and C c~ horospherical foliations. Then the geodesic flow is C ~ - t i m e preserving conjugate to the geodesic flow on a compact locally symmetric space of rank 1. REMARK. (a) Feres obtained the same result under the additional assumption that dim M -4k+2, k ~N. (b) Benoist et al. only need to assume that the foliation is C k for k ~> 2(2(dim M) 2 d i m M + 1). The optimal regularity assumption is probably C 2. This is suggested by the following result of Hamenst~idt [47]. THEOREM 2.2. Let ( M , g ) be a compact manifold of negative curvature with C 2 horospherical foliations. Then the measure theoretic entropy with respect to the Liouville measure coincides with the topological entropy. REMARK. In particular, if A. Katok's entropy rigidity conjecture holds true (see the end of Section 4.1 ) C2-regularity of the horospherical foliation would imply that the mani/bld is locally symmetric. In any case one has" COROLLARY 2.3. Let (M,g()) be a ~'ompact locally symmetric spat'e of negative curvature and g a metric, negatively ~'urved and cot!formally equivalent to g~). Then the Anosov splitting is C 2 if and only if g and g~) ~'oin~'ide up to a ~'onstanr PROOF. The proof follows from an entropy rigidity result of A. Katok (see Corollary 1.8 in Section 4). 1-7 Following Ledrappier [67] we call a negatively curved manilbld asymptotically harmonic if the mean curvature of the horospheres are constant. We recall that harmonic manifolds are those, where the density of geodesic spheres depends only on their radius or equivalently that harmonic functions have the mean value property. Clearly, harmonic manifolds of negative curvature are asymptotically harmonic. Foulon and Labourie [35] used Theorem 2.1 to prove: THEOREM 2.4. Let (M, g) be a compact asymptotically harmonic manifold of negative curvature such that the mean ~'urvature of the horo,wheres are constant. Then the geodesic flow is time preserving conjugate to a geodesic.flow on a compact locally rank 1 symmetric space. Using Theorem 3.1 of Besson, Courtois and Gallot one obtains: THEOREM 2.5. Let (M, g) be a compact manifold of negative curvature such that the horospherical foliation is C ~ or that the manifold is asymptotically harmonic. Then (M, g) is a locally symmetric rank 1 manifi~ld.
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G. Knieper
PROOF. As pointed out above for surfaces one needs only to assume that the foliation is of class C l' l to guarantee constant curvature. Therefore, we assume that the dimension of the manifold is at least 3. According to the results above the geodesic flow is C ~ conjugate to a geodesic flow on a compact locally symmetric space M0. This implies that the fundamental groups of M and M0 are isomorphic and the volume and the topological entropy of the two manifolds coincide. Hence the result follows from the result of Besson, Courtois and Gallot stated above, l--] In all proofs of the theorems stated above a certain connection or a slide variant of it plays a crucial role. It has been introduced by Kanai and is called Kanai connection. We will present a version of this connection introduced by Benoist, Foulon and Labourie. Let (M, g) be a compact n-dimensional manifold of negative curvature, 0 the natural l-form introduced in Section 1.1 and co = dO the associated symplectic structure. Assume that the Anosov splitting (see Section 1.3) T S M = E s G E II 9 E 4~
of the geodesic flow 4)t = S M ---> S M is of class C I. Then there exists a connection uniquely determined by the following three properties. (i) V0 = 0, Vco = 0 and V E" C E", V E" C E u . (ii) Denote by p" : T S M --~ E" and l;i : T S M --~ E 'l the projection onto the stable and unstable bundle. Then V x , X 'l - p " [ X " , X"]
and
V x , , X " - - p " [ X " , X"]
for all sections X 'l ~ F ( E " ) and X" 6 F ( E " ) . dlt-()4)t(v) is the infinitesimal generator of the geodesic flow and (iii) If X ( ; ( v ) X" E F ( E " ) , X " ~ F ( E " ) Vx.X"--[X(;,
X " ] - c~X '
and
Vx~. X" - - [ X ( ; , X" ] + c ~ X ",
where cr -- h t ~ / ( n - l) and ht,~ is the Liouville entropy of the geodesic flow. REMARK. (a) For c~ - - 0 this is the connection originally introduced by Kanai. (b) If I = p" - p ' the bilinear form h ( X , Y) -- w ( X , I Y ) + O ( X ) . O(Y)
defines an indefinite, nondegenerate metric on S M . Since E" and E" are Lagrangian subbundles they are isotropic with respect to h. Property (i) implies that h is parallel, i.e., Vh = 0. However, V is not the Levi-Civita connection of h, since the torsion T is nonvanishing and given by T ( X , Y) -- co(X, Y) . XG + o ( O ( X ) I ( Y )
- O(Y)I(X)).
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(c) If the Anosov splitting is of class C 2 the curvature tensor R of the connection X7 is defined. A computation shows
R ( E " , E " ) - - R ( E S , E'~)--O. In particular, the stable and unstable leaves are flat. The line bundle A t' := A "-I E u of volume forms on E" plays an important role in the rigidity theory of the Anosov foliation. Under the assumption that the Anosov splitting is C 2 the curvature of the induced connection is well defined and given by the flow invariant continuous 2-form F2" (X, Y) = tr R(X, Y)E, for X, Y ~ T S M . Using this fact, Hamenst~idt showed that under the assumption of a C 2splitting the topological and Liouville entropy coincide. For the rigidity Theorem 2.1 of Benoist, Foulon and Labourie an extra ingredient is necessary. In [43] Gromov introduced the notion of rigid geometric structure and proved under certain assumption that the structure is locally homogeneous. As a first step in their proof, the above authors show that such a structure is given by the Kanai connection, provided the Anosov splitting has a certain amount of smoothness.
4. Entropy and rigidity 4.1. Entropy comparison and rigidity Estimates which compare entropy for a pair of metrics on a given compact manifold were first given by A. Katok [64]. He applied those estimates to obtain entropy rigidity results. In particular, for surfaces of higher genus and nonpositive curvature he showed that topological and Liouville entropy never coincide, unless the metric has constant curvature (see Corollary 1.9). In this section we will derive slightly more general versions of his comparison estimates. The proves presented here followed to a large extend Katok's original ideas but have a little more of a geometric flavor. In particular, the notion of geodesic stretch which measures the asymptotic ratio of a pair of Riemannian metrics is very useful. Let M be a compact Riemannian manifold with universal covering X. Fix two Riemannian metrics go and g on M and denote their lifts to X again by go and g. For each vector v in the unit tangent bundle (SM)~ o of go consider the geodesic c,,(s) with respect to the metric go with initial condition b,,(0) = v. If g,, denotes a lift of c~, to the universal covering we define a(v, t) := de (~7,,(0), ~?,,(t)), i.e., we measure the endpoint of the geodesic segment r t]) with respect to the metric g. The triangle inequality implies that a (v, t) is a subadditive cocycle, i.e.,
a v, ,, +
a v, ,,
+ a
v, '2).
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G. Knieper
The following proposition is an easy consequence of the subadditive ergodic theorem [57,
83].
PROPOSITION 1.1. Let/Z be a Cgo-invariantprobability measure
o n ( S M ) g o.
Then
a(v,t) t---, or t
It~ ( g o , g , v) - - lim
exists for ~z-almost all v e (SM)g o. Moreover, we have Iu(go, g, v)d/z = lim t---~ o c t
M )go DEFINITION
M)~()
a(v,t)d/z.
1.2.
I~ (go, g)
"--
fSM)xo
I~ (go, g, v) d/z
is called the geodesic stretch of the metric g relative to g0. The geodesic stretch is bounded from above by the /z-average of the norm II I1,~,= g ( , )l/2 with respect to g (see [64]). More precisely: LEMMA
1.3.
f It, (g(), g) ~< / J( S M
Ilvll,~ d/z. )~(~
REMARK. Using the Cauchy-Schwarz inequality, we obtain
tf
S M ).~'o
)2/ Ilvll,edlz ~
g(v,v)d/z. S M ) x()
The inequality is strict unless g(v, v) -- go(v, v) for/z-almost all v 6 (SM)e(). If/z is a Liouville measure/Z t., we have
S M ),~()
g(v, v) d/z = vole o (M) 9n
tr'e~ g d vole().
The quantity on the right hand side is up to a constant nothing else than the energy of the identity map id" (M, g()) ~ (M, g). The entropy comparison Theorem 1.5 below is based on the following characterization of the measure entropy for an ergodic flow, due to A. Katok [56].
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497
PROPOSITION 1.4. Let (X, d) be a compact metric space with a continuous flow ~ot : X
X. Suppose lz is a ~ot invariant and ergodic Borel probability measure. Define for T > 0 the metric dT(x. y) -----max{d(qg' (x). qg' (y)) ] 0 ~< t < T}. For c~ ~ (0, 1) and 6 > 0 denote by N(T, 6, or) the minimal number of 6-balls with respect to dT which cover a set of measure >~6. Then h n ,, (~~
log N ( T , 6, c~) ~-,0
T
7"~oo
THEOREM 1.5. Let (M, go) be a compact manifold of nonpositive curvature and lz be a Borel probability measure invariant under the geodesic flow with respect to go. Then for any Riemannian metric g, we have
h(g) >~
I;, (go, g)
h;~ (go)
where h(g) denotes the volume entropy of g (see Section 2.4) REMARK. This theorem is a generalization of a result by A. Katok [56]. He proves under the stronger assumption that go is a metric of negative curvature that
ht,,p(g) >/ f~.SMI~.,, Ilpll.~ d u " hj,(g~)) holds for all metrics g and all probability measures p invariant under the geodesic flow with respect to go. It also generalizes a corresponding result in [64], where it is only proved in the case w h e n / z is the Liouville measure. However, most of the ideas in the proof are already present in Katok's original paper. PROOF. Since every invariant measure has an ergodic decomposition one can assume that p is ergodic. Let f C X be a fundamental domain in the universal covering X and let ~ be a positive number. For T > 0 consider the set
a(v,t)
Ii~ (g0, g) < e , t ~ T } .
The ergodicity of/z implies that/z(Sf).~ 0 \ A T,~:) ~ 0 as T ~ o~. Hence, we can choose T > 0 such that p(AT'~:) >~ 1/2. For t > T and 6 > 0 let S(t, 6) be a maximal subset of A T,~: which is 6-separated with respect to
d,(v. w ) = maxId~,,(c,,(s), c,,(s)) l O ~ s <~t I.
II
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oe
Hyperbolic dynamics and Riemannian geometry
499
Then the geodesic flows with respect to go a n d g are, up to a scaling, time preserving conjugate. More precisely, there exists a h o m e o m o r p h i s m F : ( S M ) g o --+ ( S M ) g such that F o dpgl,- ~; o F, where c -- h ( g ) / h ( g o ) . REMARK.
(a) This theorem is essentially due to Burger [20]. He proves that the marked length spectra of g() and g are proportional. This implies the existence of a conjugating homeomorphism. See [64] for a different proof. (b) The existence of such a conjugacy implies in dimension two that g() and g are isometric up to a constant (see Theorem 5.1 of Section 4.5). The next result follows easily from Theorem 1.5 and Lemma 1.3. COROLLARY 1.7. Let (M, g()) be a compact manifold o f nonpositive curvature a n d lZ be a dp.e() invariant probability measure. Then, f o r any other metric g the estimate
h t~ (go)
h ( g ) >~
(1.1)
f~ S'M t~,, II v I1,,~dl~ holds, l f ~ = lz 1. is the Liouville measure,
h(g)2 >~ I
vole()(M) h t, i. (g()) 2 JM tr~(~g d vol~(~
(1.2)
If additionally g is confi~rmally equivalent to g(), we have
h(g) )
( vol~() M ) I/,, vol~ M
ht'c (g())"
(1.3)
Itg() is not a flat metric inequalities (1.2) and (1.3) are strict, unless g and g() are identical up to constant. REMARK.
(a) If (M, g()) is a compact manifold of nonpositive curvature, the metric g() is nonflat iff the Liouville entropy hni (g()) > 0 is positive. This is a consequence of Pesin's entropy formula given in Section 2, Theorem 4.3. (b) It turns out that the last inequality is true for metrics not necessarily conformally equivalent to g(), provided g() is a locally symmetric metric of negative curvature (see Section 4.3). As a consequence, we obtain the following entropy rigidity result due to A. Katok [56] for a conformal class of a locally symmetric space of negative curvature.
G. Knieper
500
COROLLARY 1.8 (Katok). Let (M, go) be a locally symmetric compact manifold of negative curvature. If g is a metric of nonpositive curvature, conformally equivalent to go with volg M = volg 0 M, then
h(g) >~ht~,. (g). Furthermore, equality holds if and only if g = go. PROOF. Since go is a locally symmetric metric of negative curvature p,/~ is the measure of maximal entropy, i.e., the Bowen-Margulis measure. Using that g is conformally equivalent to go with the same total volume, we obtain from (1.3) in Corollary 1.7, h(g) ~ h(go), where equality holds if and only if g = go. Applying Corollary 1.7 again, we conclude that h(go) >~hl~ (g). [-q This yields for surfaces: COROLLARY 1.9 (Katok). Let M be a surface of a genus >~2. Then for all Riemannian metrics g ~?fnonpositive curvature
h(g) > h!~1 (g), unless g is a metric ~?['constant negative curvature. PROOF. The proof follows from the uniformization theorem which states that each Riemannian metric on M is conformally equivalent to a metric of constant negative curvature. [--I It is a well-known conjecture due to A. Katok [ 18] that Corollary 1.9 should generalize to higher dimensions in the following sense. CONJECTURE (Entropy rigidity). Let (M, g) be a compact manifold ~?[negative curvature.
Then h(g) = h / , / ( g ) , if and only if(M, g) is locally symmetric.
4.2. Regularity of topological entropy As we have seen in the last section the entropies provide interesting functionals on the space of Riemannian metrics. Those functionals are never determined by the local geometry of the manifolds unless we are in special symmetric situations. The next theorem shows that the topological entropy is smooth on a certain class of nonpositively curved metrics. Originally this has been proved in [59] for negatively curved metrics. However, the
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501
theorem remains true for nonpositively curved metrics provided the measure of maximal entropy is unique. As we will see in Section 5, the uniqueness holds for rank 1 manifolds, i.e., manifolds of nonpositive curvature with some amount of hyperbolicity. THEOREM 2.1. Let (M, g0) be a compact rank 1 manifold and g z , - e <~ )~ <~ e, a perturbation within the space of metrics of nonpositive curvature. Then )~ e--> h(gz) is a C I -function and its derivative is given by d d)~
h(g())
h (g~) --
2
f,s'
M)~,)
d
dJk
gz (v, v) d/~g,,,
where/~() is the measure of maximal entropy f o r the geodesic flow 4)~(). PROOF. Using Corollary 1.7, we obtain
1 h(g())
./ISM)x( ' Ilvll~ dlz~,,
- 1) ~< h ( g z ) -
h(g())
IIv I1~,, d/z e>. - 1) .
Linearizing )~ ---> IIv ll,~,>, at ) ~ - 0 implies
II v I1,> - II v I1,,, + 9~
d
Ilvll.~, + o(~, v). s =()
Substituting this in the above inequality, we obtain for suitable o t(~), o2(~.)"
( f
h (g()) -;k
d
SM)~,,, ds
Ilvll,~, dlzo + ol (~.)) s =()
<~ h(gx) - h(g()) <~ h(g()) -)~
,S'M)~;,
ds
II vtl,~, dlz,~; + o2 ( k ) ) . s =()
Each weak limit of (lZq,~) for X -+ 0 will determine a measure of maximal entropy for g(). Since this measure is unique in the rank 1 situation (see Section 5.5) X --~ #,~>, is continuous in the weak topology. Consequently, we obtain the desired formula for the derivative. F1 Using symbolic dynamics the authors prove in [58]" THEOREM 2.2. Let (M, g()) be a compact manifold with a C~-metric of negative curvature. Let X ---> g ~ , - ~ <~ )~ <<.e be a C~-perturbation then the function )~ e--> h(g~) is a C ~-function as well.
G. Knieper
502
The result has been extended to the Liouville entropy and even more generally to pressure by Contreras [22].
4.3. Minimal entropy A celebrated theorem by Besson, Courtois, and Gallot [10] states that inequality (1.3) in Corollary 1.7 remains true for all metrics g provided go is a locally symmetric metric of negative curvature on a compact manifold. In fact, it even holds if g is a metric on a manifold with fundamental group isomorphic to the fundamental group of the locally symmetric space. We call this theorem, which was also conjectured by Gromov [41], the minimal entropy theorem. THEOREM 3.1. Let (M0, go) be a compact locally symmetric space of negative curvature. Assume that (M, g) is another compact Riemannian manifold such that rcl (M) ~- rcl (Mo). Then (vol~,, M) !/'' h(g) >~ (vole M)l/'' h(go). /f dim M~)= dim M ~> 3 equality holds if and only if g and g~) are homothetic, i.e., are isometric up to a constant. The rigidity part of this theorem immediately implies the Mostow rigidity theorem for compact locally symmetric spaces of negative curvature. COROLLARY 3.2. Let (Mo, gl)) and (Mi, gl ) be two compact locally symmetric spaces of negative curvature and of dimension strictly bigger than 2. Then gl and g() are homothetic provided the fundamental groups of Mo and MI are isomorphic. Now we will give a proof of Theorem 3. l under the assumption that (M, g) is a compact manifold of negative curvature. In this case, the use of the Patterson-Sullivan measure constructed in Section 2.5 simplifies the proof considerably (see [ 1 1]).
Step 1 (Construction of the homeomorphism between the ideal boundaries). Let M0 = (X/Fo, go) and M = (Y/F, g) be compact homotopy equivalent manifolds of negative curvature. Then the homotopy equivalence q9 : M --+ M0 lifts to a quasi-isometry ~ : Y --~ X which is equivariant with respect to the F-action on Y, i.e., ~ov
= A(V) oq3
for all V 6/-', where A : F -+ /-]) is an isomorphism between the fundamental groups F of M and/]) of M0. Furthermore, ~ induces a homeomorphism F : Y ( ~ ) --~ X ( ~ ) between
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503
the ideal boundaries of Y and X which is equivariant with respect to the F-action on Y, i.e., Foy=A(y)
oF.
PROOF. It is a general fact that a homotopy equivalence between compact manifolds lifts to a quasi-isometry between the corresponding universal coverings (see 2.10). Furthermore, Corollary 2.9 of Section 2 implies that the quasi-isometry 99 : Y --~ X induces a homeomorphism F : Y (cx~) ~ X (cxz) between the ideal boundary. The /-'-equivariance of the quasi-isometry implies the/-'-equivariance of F. IN Step 2 (Construction o f the natural map). In Step 2 we construct, using the barycenter of the Patterson-Sullivan measure, a differentiable/-'-equivariant extension f : Y -+ X of the homeomorphism F : Y ( ~ ) ~ X ( ~ ) . The /-'-equivariance of f yields that f can be viewed as a map from M to M0.
LEMMA 3.3 (Barycenter of a measure). Let X be a Hadamard manifold o f negative curvature and let xo E X be a fixed reference point. Then, f o r any finite measure )~ on X ( ~ ) , the function q : X ~ IK given by
q(x)-
fx
(,,x:)
bx,,(x, ~) dX(~)
is smooth and depends only up to an additive constant on xl~. Moreover, tfsupp ,k = X (cx:~) then q is strictly conv~:~ and there e~ists a uniquely determined absolute minimum S = S()~) E X which is called the bao'center o f'/k.
PROOF. The smoothness of q follows from the smoothness of the Busemann function x --~ bx~,(x, ~). Furthermore, the equation b~,(x, ~) - b,~ (x, ~) = b~,,(xi, ~) implies that q only changes by an additive constant if we choose another reference point X l. It is a consequence of the convexity of the Busemann function that q is strictly convex if supp)~ = X(cx~). Using again that ,k has full support, we obtain that q ( x ) --~ cx~ as x --+ ~1 E X(~:~). Hence, q has a unique minimum. [-1 LEMMA 3.4. Let M = ( Y / f ' , g ) and M~ = ( X / ~ ) , g o ) be two compact homotopy equivalent manifolds o f negative curvature and F : Y ( c x ~ ) ~ X ( ~ ) be the induced homeomorphisms between the ideal boundaries. Let {lZ v } vE Y be the Patterson-Sullivan measure o f Y and S be the barycenter-map introduced in Lemma 3.3. Then the map f : Y ~ X given by .
.
f (y) = S ( F . l z v ) is a smooth F-equivariantfunction, i.e., f o y ( y ) = A ( y ) o f (y) f o r a l l y E Y and y E F.
REMARK. Following Besson, Courtois and Gallot [ 11] we call f the natural map.
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504
PROOF. Since for each y e Y f (y) e X is the unique critical point of the map
g"(x ' F(~)) d/~y (~) Xo 0 , ~')d(F,#y)(~') -- fy q(x)= fx(oo) X~:~ (oo) it is for each y e Y the unique solution of the equation O = G(x , y)"= fx
gradt,t"X() g"(x , F ( $ ) ) d # ~' (~) 9 (oo)
Furthermore, the map G ' X x Y --+ T X is differentiable and for each fixed y e Y the covariant derivative of the vector field x ~ G(x, y) on X induces a nondegenerate linear map Vg~ y)" TxX ~ TxX given by t,,g0(x , F ($)) d# v($) V,~'G(x, y)= fx(~) V~'}' grad u.r() Hence, the implicit function theorem is applicable and implies the smoothness of f . The F-equivariance follows from the F'-equivariance of the Patterson-Sullivan measure and the fact that the barycenter-map commutes with isometries. More precisely, for all y e F' and y ~ Y we have:
f (y y) -- S(F,/t•
= S( F, y, pt ,.) = S( F o y),/t,.
= S(A(y),F,#>.)- A(y)S(F, lt,.)- A(y)f(y).
i-1
Step 3 (The derivative of the natural map).
As usual, the implicit function theorem also provides a formula for the derivative of the implicitly defined map. To simplify the notation in our computation, we denote for any Hadamard manifold (Z, h) the gradient grad b~l,,(z, r/) of the Busemann function b h:0(z, r/) by Bh(z, 77) (see the remark in Section 2.3). Now, fix a point y0 e Y. Using the transformation rule of the Patterson-Sullivan measure, it follows that the natural map f : Y --+ X fulfills the equation
O-- fx
( ,"~ )
B'~"(f (y), F($))e -hl'~l'.'''l'''~ du,.,,($).
In order to compute the derivative of f , we choose a differentiable curve t ~ y(t) e Y such that y(0) -- y and #(0) -- v. Then, differentiation of the equation above yields" D dt
B.~"(f(y(t)), F($))e -h/~)h:'~',,(-''(')'s) d/t,.,, ($) t --() i X (~'x~)
B.~"(f(y), DI'( v)v -- fx ( ~ ) V'U~ 9
F($)) dkt,.($)
.
- h(g) f x ( ~ )
B.~~
F(~))(B,~(y,$), v)d#,,(~).
(3.1)
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505
i be the normalised Patterson-Sullivan measure. Consider for Let /.iv = ~,,,(x(oc))/Zy x 9 X and y 9 Y the endomorphism Qx,y: Tv X --+ Tv X given by
Qx,y(W) - f x
V~'l'Bg('(x, (ec)
F(~)) d/2,,(~)
(3.2)
and the linear map
Lx,v : T r Y ---->T r X given by
Lr.y(V)
-
fx(o~)Bg~
-
F(~))(B'~(y,~),
v)d/2r(~).
Then Equation (3. l) can be written in the following compact way: (3.3)
Q.f(y),,, o D f ( y ) = h(g)Lf(.,,),y.
Denote by U(,.,t) the second fundamental form of the horosphere corre ~ponding to x E X and 71 E X ( ~ ) . Then for all u~ E T~ X, we have
V',~?B'~""(x, F(~))= U(,.t.(~,,(u,-
(u,, B'V"(x, F(~')))B'V"(x, F(~))).
Since U(.~-.F(,~)) is symmetric and positive definite, the map Q.,-,~.: T~.X ~ symmetric and positive definite as well and, therefore,
D r ( y ) -- h(g)QTl,.)..,, o
L i(.,.i.,..
T~-X is
(3.4)
Step 4 (Estimate o f the Jacobian of the natural map). Let w,vo and to v be the volume forms associated to the Riemannian metrics g(I and g. Define the Jacobian of .f to be Jac f ( y ) = det D f ( y ) , where the determinant of a linear map is defined via the volume forms. Therefore, Equation (3.4) implies Jac f ( y ) = h" (g) LEMMA 3.5. For all x
det L.I. ( v ). v det Q f(y i. v
9X and y EY, we have
I det L.,.,.,. I ~< ~
1
(det H,.. ,.) I
/2
where Hv, r : Tv X --+ Tv X is the symmetric endomorphism given by
Hv.y(W)
"
-
-
fx(~)(,,,, 8.,,, (x,
8.",, (x,
506
G. Knieper
REMARK. It would be of considerable interest to know under which assumptions I det Lx,yl is bounded from below by a positive constant. In general, due to the existence of exotic structure, we cannot expect the existence of such a constant. PROOF.
The Cauchy-Schwarz inequality implies for all w 6 Tr X and v 6 Tv Y
I(Lx,xV,
w),e,, I -
fx(~) (Bg"(x, F(~)), w)
<~(Hr,y(W), W)'/2 fx
(ec)
v)d/iv(~)
(B'e(y,~), v)2d[zv(~).
(3.5)
Now choose an orthonormal basis el . . . . . e,, which diagonalizes the symmetric endomorphism Hr,v :TrX --+ TrX. Then H x , y ( e i ) - - l z i e i , and since IlB,e~ F(~))llgo = 1 and/2v is a probability measure, the definition of H~-,v implies ,tZ i > 0 and Y'/i~l/zi = 1. Since the lemma is trivially true if det L r, r = 0, we assume det Lx, v ~ 0. In this case, Lx,y is invertible, and we can apply the G r a m - S c h m i d t orthonormalization process to the basis L -.~,y I (e j). If bl . . . . . b, is the resulting orthonormal basis in Tv Y, we have .J
L.v,v(bj) -- Z cijej, i=1
i.e., the associated matrix is upper triangular. Using (3.5),
ej)l
IdetL.,.,.I.j= I
~<
lZl (H,.v(ej) , e.i) I/2
.j = I
(B'e(y,~),bj .j = I
d/~v(~)
(~)
1
Since the geometric mean is bounded from above by the arithmetic mean, we obtain
]det Lx' xv l <~ (det" H~""v ) l /2 ( -nl f i f j =I = (detH~. v)l/2
(B g (y, ~), bj) 2 d/~ v(~-)) ''/2
1
nn/2 9
I-1
This implies the following estimate for the Jacobian of f :
[Jac f(Y)l ~< h"(g)
(det Hf (y), y) I/2 1 det Q.f(y),v " n"/2"
(3.6)
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Step 5 (Proof o f the minimal entropy theorem). Only in the last step we will assume that the target manifold (M0, g0) is a locally symmetric space of negative curvature. To simplify the calculation we will assume that (M0, go) is a manifold of constant curvature - 1. Then the second fundamental form of the horosphere is the identity and the endomorphism Qx.y :Tx X ~ Tx X defined in (3.2) is given by Qx. v(w) = w - H~. r(w). I f / z l . . . . . /z,, are the eigenvalues of Hf~y).,., we obtain, using (3.6), the following estimate for the Jacobian of f :
IJacf(Y)l ~
1-I.';= I /zl/2 ./ h"(g) l-I.'~=l (1 - / z j )
9
1
(3.7)
nn/2 9
So far, we did not make use of the assumption n ~> 3. Since for surfaces of higher genus the Teichmtiller space of nonisometric metrics of constant negative curvature is nontrivial, the rigidity part of the minimal entropy theorem cannot be true for n -- 2. It is an amusing fact that the assumption n ~> 3 will be only used in the following innocent looking lemma (see [10]). LEMMA 3.6. Let n be an integer strictly bigger than 2. Let lzi . . . . . lz, be real numbers such that 0 < lz i < 1 and }--~.'i=_I/zj - 1. Then i-iit.i=i/zj!/2
( l / n ),/.,-
<~
H
l-I i = l ( 1 - / t j
)
( 1 - 1/n)"
where the inequality is stri~'t unless I~l -- 1~2 . . . . .
I~,, -- l / n .
The minimal entropy theorem will be a consequence of the following proposition. PROPOSITION 3.7. Let (MI), gr be a compact n ~ 3-dimensional symmetric space o f negative curvature and ( M , g) be any compact manifold o f negative curvature homotopy equivalent to Mo. Then, f o r all y ~ M, the following estimate f o r the Jacobian o f the natural map f : M --+ M~) holds: [Jac f ( y ) [ ~<
h"(g) h" (gt))
Furthermore, this estimate is strict unless I,~go) D f (y) is an isometry.
PROOF. We will prove this proposition under the assumption that (Mo, go) has constant curvature - 1. The inequality is a consequence of Lemma 3.6 since
IJacf(y)l ~< and h (g0) -- n - 1.
l--I';= I I~.jt /2
1
h" (g) i--l,}=, ( 1 _ lt.i ) n"/2
<~
h"(g)
h"(g)
(n - 1)"
h" (go)
0,.,~
~ ~ ~.,-i
.wl
;;~
Z ~
ov,-i
9
i
~,.,."
I
~
II
II
I II
=:
9
......,
II
E
m
;
m
.'-~
t-.
~
~
~
r162
%//
~
~..,
t ~176
......
II
~
~ ,.,,q
o
~
"~
,..,
,d
.~
~
=
Z
,...,
~-
,,I I
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509
where L := Lf(y),y, we obtain from (3.8) by choosing an orthonormal basis bl . . . . . b,, Tv Y I1
t r A - - n 2 h2(g) t r L t L - - n 2 h2(g) E ( L b j , h 2(go) h 2(go) j= I ~< n2 h2(g) 1 ~ h2(go) n
j= I
fr"
Lbj)
h2(g) (BZ (y' ~)' bj)2 d/2v(~)= n ~ h2(go) (~)
Hence, (tr--~A)"~<(h2(g)) ' ' - h2(g(,) n
h2"(g) h 2, (go)
= det A
and, therefore, h2 (g) id.
A - D f (y)' . D f ( y) = h2(g() ) This implies h2(g) --(v.
h2(g~)
v).~- - ( D r ( y ) ' D f ( y)v, v ) - (Df ( y)v, D.f ( y)v).~,, ,
i.e., ~hI'e~D.f(.v) is an isometry.
I-q
Now, the minimal entropy theorem follows quite easily from Proposition 3.7. Namely, since f has degree 1, we obtain
vol(M(),go)--fM u'.e,,--fM f* wg. -- fM Jac.f (y) u,z 1)
h" (g) fM h" (g) vol(M g) <~ h" (gl~------~ u,e -- h" (gr ' " The inequality is strict, unless Jacf(y) -- h" (g)/h" (g(~) for all y E M. In this case f is an homothety. QUESTION. Does the above estimate remain true if (M, go) is a compact locally symmetric space of nonpositive curvature and higher rank? 4.4. Spectral rigidity Let (M, g) be a compact Riemannian manifold and denote by Ag "C~(M) --->C ~ ( M ) the Laplace-Beltrami operator with respect to the metric g. A smooth perturbation gx, - s ~<
510
G. Knieper
~< e of the metric g = go is called isospectral if the spectra of A gx and A g o coincide for all ~ 9 l - e , +e]. A Riemannian manifold (M, g) is called spectrally rigid if all isospectral perturbations gz of g are trivial, i.e., gz = qg~g, where ~0z'M ~ M is a one parameter family of diffeomorphisms. The following theorem, due to Croke and Sharafutdinov [25], generalizes previous work by Guillemin and Kazhdan [44,45], and Min-Oo [73]. THEOREM 4.1. A compact Riemannian manifold o f negative curvature is spectrally rigid. The first step of the proof is that the Ag-spectrum of a compact negatively curved manifold ( M , g ) determines the length spectrum A ( g ) C Ii~+ of periodic geodesics (without multiplicity) on M. This is a simple consequence of the trace formulas due to Duistermaat and Guillemin. Let s be the set of free homotopy classes of M. If g is a metric of negative curvature we call the map Lg :S-2 ~ R + which associates to each free homotopy class the length of the unique closed geodesic representing this class, the marked length spectrum. COROLLARY 4.2. Let g z , - e <~ X ~ e, be an isospectral perturbation o f a negatively curved metric g = go on a compact manifold. Then Lg~ = Lg o fi~r all )~ 9 l - e , +e]. PROOF. This follows from the continuity of)~ ~ L ez (c~) and since L ez (a) is contained in the countable set A(go) C IR+. Hence, L ~ (c~) is constant for all c~ 9 s [-] REMARK. In particular, for surfaces the theorem follows from the work of Croke [23] and Otal [751 (see Section 4.5). Theorem 4.1 is now a consequence of the rigidity of the length spectrum. THEOREM 4.3. Let (M, g) be a compact manifold o f negative curvature and g x , - e ,k <~ ~, be a perturbation such that A ( g z ) = A(go). Then the perturbation is trivial.
<~
The first step in the proof of this theorem consists of the following simple observation: LEMMA 4.4. Let gz be a perturbation as above and fl -- if2 [z=z~gz. Then fi~r each closed geodesic c:[0, ~1 ~ M in (M, gz~), we have
Using this simple lemma, there are two main ingredients for the proof of Theorem 4.3. The first one is due to Liv~ic [68]. The theorem asserts that to each function f 9 C ~ ( S M ) which integrates to zero along each periodic trajectory there exists a function F : SM ~ II~ such that X~; F = f , where X c ( v ) = (v, 0) is the infinitesimal generator of the geodesic flow (see Section 1.1) (the smooth version of this theorem is due to de la Llave [69] et al.). The second main ingredient which distinguishes the work of Croke and Sharuftdinov from previous approaches is a certain formula of Weitzenb6ck type on the tangent bundle
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of a manifold. This formula has been discovered by Pestov and Sharafutdinov [8 1] and is called Pestov's identity. We will give an coordinate free presentation of this formula (see the Appendix for a proof). To state this formula, we introduce the following notations. If q9 9 C e~ ( T M ) , we denote by gradqg(v) -- (grad h qg(v), grad v qg(v)) 9 T v T M the gradient of 99 with respect to the Sasaki metric. The vector fields grad h 99(v), grad v qg(v) 9 TjrI,,)M are the associated horizontal and vertical components. A smooth map X : T M ~ T M is called a semibasic vector field if it preserves the fibers of T M, i.e., n" o X = n'. The horizontal and vertical gradient are examples of such maps. For a semibasic vector field there is a natural notion of the horizontal and vertical divergence, div h and div v (see Appendix). THEOREM 4.5 (Pestov's identity). For all ~o e C ~ ( T M ) , 2(grad h 99(v), g r a d V ( X ~ ; q ) ( v ) ) ) -
[Igrad"
we have
II2 + d i v h y ( v ) + divVZ(v)
- ( R (grad v qg(v), v) v, grad" q9( v )), where Y(v)-
(grad h 99(v), gradVqg)v - ( v , grad h qg(v)) grad v g)(v)
and Z ( v ) -- X~;qg(v) 9grad h 99(v) --(v, grad h 99(v))grad h qg(v).
Let C , , ~ ( T M ) -- {q9 E C ~ ( T M ) I qg(~.v) -- )~'"qg(v), for all ~. > 0} the homogeneous functions of degree m. Then we obtain the following integrated version of Pestov's identity. THEOREM 4.6. Let M be a compact n-dimensional Riemannian man~['old and q9 E C,,~ ( T M ) . Then
- 2 L M ~Pdivh gradV (X~;qg) dlz t.
-- LM ]]gradh ~p(v)II2 d/tt.
+ (n + 2m) f (XG~p(v)) 2 d/,L
-- L'M (R (gradV qg(v), v)v,
grad v ~o(v))d#t.
G. Knieper
512
REMARK. This formula, together with Liv~ic's theorem, immediately shows: If M is a compact manifold of negative curvature and f 6 C ~ ( M , R) satisfies fc f o zr dt -- 0 for all closed geodesics c, then f = 0. Namely, if f -- XG~0 for a function ~o 6 C ~ ( S M , Ii~), the left hand side of the identity above is zero since grad v f -- 0. But then f -- Xcq9 = 0. Now we want to generalize this to symmetric m-tensor fields. Denote by F m ( M ) :-- F ( Q m T * M ) the C ~ - s e c t i o n s in the m times covariant tensor bundle (m-tensor fields) and S m ( M ) = F ( @ ~ y m T * M ) be the subbundle of symmetric tensor fields. Denote by V : F m (M) --+/-,,,,+ l (M) the covariant derivative, induced by the Levi-Civita connection V and given by (Vc~)(X, Xl . . . . . Xm) = (Vxc~)(Xl . . . . . Xm) I11
= X ~ ( X i . . . . . Xm) - ~
~ ( X l . . . . . X i - I , V x Xi, Xi+l . . . . . X,n)
i=1
for vector fields X, Xi
7
9 S'"
M and c~ E F'" (M). If we restrict V to S'" (M), we obtain
on
m * M) . ( M ) ---> F ( T * M | 6~ ._.~ymT
Now we compose V with the symmetrization cr 9 F ( T * M @ .-.sym ~'" T ' M ) --+ S '''+l ( M ) and obtain an operator
* : S'" ( M ) ~
S'" + I (M).
The Riemannian metric g induces an inner product ( , ) integration induces an inner product ( , ) on the sections
on the fibers of |
and
(c~, fl) -- f M (C~, ~) d vol. The adjoint of g* with respect to ( , ) given by got = -tr1,2
is the negative divergence g : S '''+l (M) --+ S ''z (M)
Vc~.
In particular, if Ei . . . . . E,, are orthonormal vector fields we have for all vector fields X I . . . . . Xm that I1
Sc~(XI .....
X,,,) =
-
~-'~(VEiot)(Ei, Xl . . . . . X,,,). i=1
One obtains the following decomposition of symmetric tensor fields.
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513
PROPOSITION 4.7. Let ( M , g) be a c o m p a c t R i e m a n n i a n manifold a n d ot e S m ( M ) be a C~ o f the symmetric tensor bundle. Then there exist sections fl ~ S m - ! ( M ) , g ~ S m ( M ) such that 8y = 0 a n d c~ = 6 * f l + y.
This decomposition is unique if m is even. I f m is o d d it is unique provided (fl ' | e where g is the R i e m a n n i a n metric and ~ = (m - 1)/2.
= 0,
REMARK. The restriction for odd m is necessary since V ( | eg) -- 0 by the parallelity of g. To apply Pestov's identity in the case of symmetric m-tensor field one needs the following observation. LEMMA 4.8. Let o t e S'" ( M ) be a symmetric tensor field a n d F ( v ) = a ( v . . . . . v). Then, ]'or all v e T M,
m6ot(v . . . . . v) -- - d i v h grad v F ( v ) and X(; F ( v ) -- 5 * a ( v . . . . . v). PROOF.
d
(grad v F (v), 77) --
ot(v + tO . . . . .
v + tO)
-
-
mot(o, v .....
v).
1=(}
Note, that for a semibasic vector field X the horizontal divergence is given by
div h X ( v ) --
(X (V,,i (t)),
7
e.j ),
t--l)
j= I
where ei . . . . . e,, is an orthonormal basis of T~rl,,)M and v,,j (t) is the parallel translation of v along the geodesic c,.i. Hence,
- d i v h grad v F ( v ) -- m
kd
-~
j = I
ot((ei (t), v,,i (t) . . . . . v,,i (t)) l
I!
=
m Z ( V e i c ~ ) ( e . / , v . . . . . v) ./= I
=
m 9~u.
The second equality follows since d
X~; F ( v ) -- -~
u(r t--O
.....
r
v) -
(v~,u)(v .....
v) - a * u ( v . . . . .
v).
I-1
G. Knieper
514
Now we can prove the following theorem due to Croke and Sharafutdinov [25]. THEOREM 4.9. Let (M, g) be a c o m p a c t negatively c u r v e d R i e m a n n i a n m a n i f o l d a n d ot ~ S m M a s y m m e t r i c tensor field. I f the integral
fo
e a ( b ( t ) . . . . . b(t)) dt
is equal to zero f o r all closed geodesics c" [0, g.] --+ M, there exists a s y m m e t r i c tensor f i e l d fl ~ S m - I ( M ) such that (~ * ~ ~
Ol.
I f m is even, fl is uniquely determined.
PROOF. The smooth version of Liv~ic's theorem implies the existence of a smooth function qg : S M ---> R such that XGcp(v) = a ( v . . . . . v). Extending r to a function on T M such that r 6 C ~ ( T M ) is homogeneous of degree m - 1, we have XGcp(v) =c~(v . . . . . v) for all v ~ T M . Decomposing c~ according to Proposition 4.7, we obtain: X(;q9 = ~ = •*/3 + y. By L e m m a 4.8 6*fl(v . . . . . v) = X ( , f l ( v . . . . . v) and, therefore, X(, (r - / ~ ) = y. By the same lemma div h grad v X(;(q9 - [4) - div h grad v y -- - m S y -- 0. Using the integrated version of Pestov's identity we deduce X(; (99 - / 4 ) - - 0 which yields the claim. F1 To prove the spectral rigidity we need to consider the special case of symmetric two forms.
COROLLARY 4.1 0. Let M be a c o m p a c t m a n i f o l d a n d )7 ~ S 2 ( M ) be such that )1 integrates to zero along all closed geodesics. Then there exists a smooth o n e - p a r a m e t e r f a m i l y o f d i f f e o m o r p h i s m s r " M ---> M such that d )1-- -~
~0;g. l--()
PROOF. By the above theorem there exists a 1-form/4 such that r/-- 6*[3. Let Z be the vector field dual to 2/4. Then one obtains for all vector fields X, Y on M d
r
~ * ~ ( X , Y) -- g ( V x Z, Y) -~- g ( V y X, Z) -- -~
Y),
t =0
where qgt is the one parameter subgroup of the diffeomorphism group associated to Z.
VI
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515
4.5. Rigidity o f the marked length spectrum In [18] it is conjectured that the marked length spectrum is rigid within the class of compact negatively curved manifolds, i.e., two metrics of negative curvature on a given compact manifold are isometric if their marked length spectra coincide. According to the last section, the infinitesimal version of this conjecture is true. For surfaces the conjecture has been proved independently by Croke [23] and Otal [75] using different methods. THEOREM 5.1. Let M be a compact surface o f genus ~ 2. If g l and g2 are metrics o f negative curvature with the same marked length spectrum, then g l and g2 are isometric. REMARK. This result has been extended by Croke, Fathi and Feldman [24] to the case of compact surfaces of nonpositive curvature. The above conjecture can be reduced to the question whether a C~ preserving conjugacy between geodesic flows is induced by an isometry [46] (see also [64]). More precisely" PROPOSITION 5.2. Let gl and g2 be metrics o f negative curvature on a compact manifold. Then the marked length spectra o f g i and g2 coincide if and only ~f the geodesic" flow o f the metrics g l and g2 are C~)-time preserving c'onjugate. REMARK. Note, that a time preserving C~
(SM)~ and ( S M ) 2 is a homeomorphism F ' ( S M ) I - - >
of the geodesic flows 4~tR,4~ on ( S M ) 2 such that 4)~ o F - - F o 4)I~.
5. Ergodic properties of weakly hyperbolic spaces In this chapter we summarize some of the recent results on the dynamics and asymptotic geometry of nonpositively curved spaces (see [65] and [66]). We are in particular interested in spaces with a certain amount of hyperbolicity which we will call weakly hyperbolic or rank 1 spaces. The rank of a nonpositively curved manifold has been introduced by Ballmann, Brin and Eberlein [7]. It measures the amount of flatness of a manifold M of nonpositive curvature.
5.1. Definition o f rank and rank rigidity DEFINITION l . l . For a unit tangent vector v ~ S M , rank(v) is the dimension parallel Jacobi fields along the geodesic c,,. The minimum of rank(v) over all v ~ called the rank of M. If rank(v) = rank(M) the geodesic c,, and the corresponding v = b,,(0) are called regular. The set of regular vectors v ~ S M is called the regular
of the S M is vector set.
REMARK. (a) Since s is always a parallel Jacobi field and since the dimension of parallel fields is not bigger than dim M we have 1 <~ rank(M) ~< dim M.
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516
(b) A parallel perpendicular field along a geodesic c is a Jacobi field if and only if the sectional curvature of the planes spanned by E ( t ) and b(t) is zero for all t 6 I~. (c) It is easy to see that rank (Mi x M2) -- rank (Mi) + rank (M2). (d) For symmetric spaces X of nonpositive curvature the rank is defined as the maximal dimension of a totally geodesic flat subspace. In the case of symmetric spaces both notions coincide since each geodesic is contained in a maximal flat. If the geodesic cv is contained in precisely one maximal flat F it follows that rank(v) - - d i m F = rank(M). In particular, c,, is regular. A fundamental result in the theory of manifolds of nonpositive curvature is the rank rigidity result of Ballmann [5] and Burns and Spatzier [1 9]. THEOREM 1.2. Let M be a compact manifold o f nonpositive curvature with irreducible universal covering X. Then the manifold either has rank 1 or X is a symmetric space o f higher rank. Theorem 1.2 implies that most compact manifolds of nonpositive curvature are rank l manifolds. We note that they do in general not admit a second metric of negative curvature. There are compact rank 1 manifolds which have Z k, k ~> 2, as a subgroup of their fundamental group. By Preissmann's theorem this excludes the existence of a metric of negative curvature. Nevertheless, we will see that the geodesic flow of such manifolds still exhibits some of the dynamical properties similar to those which are well-known in negative curvature, or more generally for Anosov flows. Therefore, we call rank 1 manifolds also weakly hyperbolic spaces. Unlike for Hadamard manifolds of negative curvature, two points at infinity cannot always be connected by a geodesic. However, in the neighborhood of rank 1 geodesics the geometry behaves very much like in negative curvature. For n 6 1~1we denote by d,, the metric on the unit tangent bundle SX defined by d,,(v, w ) = max{d(c,,(t), c,,,(t)) I t c [0, n]}. Then we have: LEMMA 1.3. Let X be a Hadamard manifold and c be a rank 1 geodesic on X. For each e > 0 there are neighborhoods U o f c ( - o o ) and V o f c ( + o o ) such that f o r all ~ ~ U and ~ V there exists a rank 1 geodesic h connecting ~ and 71 such that dl (i'(0),/~(0)) ~< e. In particular, according to the flat strip theorem, h is uniquely determined up to parametrization. Furthermore, axial isometries fixing a rank 1 geodesic behave like in negative curvature. By definition, an isometry y 6 Iso(X) is called axial if there exist a geodesic (axis) c and to > 0 such that y ( c ( t ) ) = c(t + to) for all t 6 It~. If X / F is compact all deck transformations y 6 F are axial.
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LEMMA 1.4. Let X be a Hadamard manifold and c" IR ~ X be a rank 1 axis o f y 9 Iso(X). Then f o r all neighborhoods U o f c ( - e c ) and V o f c ( + e ~ ) there exists no 9 N such that f o r all n >~ no m
V" ( X \ U) C V
D
and
y - " ( X \ V) C U
f o r all n ~ no. REMARK. Both lemmata imply that the endpoint of a rank 1 axis can be connected to any other point in X (oo) by a rank 1 geodesic. There are many rank 1 axes if X / F is a rank 1 manifold of finite volume. LEMMA 1.5. Let X be a rank 1 Hadamard manifold such that X/I-" has finite volume. If c" N ~ X is a rank 1 geodesic, then there exists a sequence o f rank 1 axes c, such that Cn ~
C.
PROOF. First one observes that in the case of finite volume the geodesic flow is nonwandering (Poincar6 recurrence). This implies that the endpoints of a geodesic c" R X are F-dual, i.e., there exists a sequence y,, 9 F such that for one p 9 X (hence, for all p 9 X) y,, p ~ c(+c ( - o o ) . Then one shows that the closure of sufficiently small neighborhoods about c ( - o o ) and c ( + o o ) are mapped into themselves by y, and y, I Since one can assume that those neighborhoods are homeomorphic to a closed bali the fixpoint theorem of Brouwer yields the claim. 71 REMARK. Under the above assumption one can even prove that the regular set, i.e., the set of rank I vectors is dense. Hence, the rank 1 axes are dense and, therefore, closed geodesics are dense on S X / F . From that one easily concludes that F acts minimally on X (cx~). More precisely" COROLLARY !.6. Under the assumptions o f Lemma 1.5 we have fi~r all ~ 9 X ( ~ )
F~ - x (oo). PROOF. If 71 9 X(oo), 71 # ~, and V C X(oo) is a neighborhood of 71, we can choose a rank 1 axis c such that c ( + o o ) 9 V and c ( - o o ) r {71,~}. Choose a neighborhood U C X (o~) of c ( - o o ) which does not contain ~. If y 9 F is the axial isometry associated to c, Lemma 1.4 implies the existence o f n 9 N such that y" (X (cx~) \ U) C V. In particular, ?,"~ 9 V. V1 In Definition 5.4 of Section 2.5 we introduced certain families of finite Borel measures {/z/,}/,cx on X (oo) called Busemann densities. They play an important role to understand asymptotic properties of the geometry and dynamical properties of the geodesic flow. In the next section we will study those measures for rank 1 manifolds. It turns out that they are uniquely determined [65].
518
G. Knieper
5.2. Uniqueness o f the Busemann density First we will show that on rank 1 manifolds X with compact quotient M -- X / F a family {#p}pEX of Busemann densities has certain semi-local properties. For ~ 6 X(c~) and p 6 X consider the projections prs " X ~ X (oo)
and
prp" X \ {p} ~ X (oo)
along geodesics emanating from se and p, respectively. Hence, pr~(x) -- C~,x (cx~), where c~,c is the geodesic with c ~ , x ( - c ~ ) = ~, c~,x(O) -- x and p r p ( x ) = cp,x(cx~), where cp,x is the geodesic with cp,x(O) = p, Cp,x(d(p, x)) = x. PROPOSITION 2.1. Let {lZp }pEX be an or-dimensional Busemann density. (a) Then there are constants R > O, g. > 0 such that lzp(prx B(p, p)) ~ g f o r all x ~ X (oQ) U X, where B ( p , p) is the open geodesic' ball o f radius p >~ R about p. (b) Furthermore, there is a constant b - b(p) such that f o r all x E X and ~ =
,:'/,..,-( - o e ) . -]e -ud(p,.v) <~ IJp(pr~(B(x, p ) ) ) <~h e -"'i(t''') b
(c) A similar estimate holds 0" we project from p E X, namely there is a constant a - a (p) > 0 such that 1 -e a
-~"'P'-') <
PROOF. The estimates in (b) and (c) follow from (a) and the defining properties o f / z p . Namely, if A C X (cx~)
~ p ( A ) = fA e-~'t"(P"l) d/zx(r/).
If A = pr~ B(x, p), or p r p B ( x , p) then I(bx (p, r/) - d ( p , x))l is bounded by a constant for all 77 E A. This yields (b) and (c). The first estimate is a consequence of the following steps. STEP 1. supp/zp -- X (oo) f o r one and, hence, f o r all p E X since I" acts minimally on x (oo). STEP 2. For each ~ E X ( o o ) and x E X there exists r > 0 such t h a t p r ~ ( B ( x , r)) contains some open sets.
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519
To see that, choose a rank 1 axis c on X. By the remark following L e m m a 1.4 the endpoints of such a geodesic can be connected to all other points of X(cx~). For 6 X ( o c ) let he., 7 be a connecting geodesic with h~.,l(+oo) = r / = c ( + o o ) . Since he., 1 is a rank 1 geodesic, L e m m a 1.4 implies that for E > 0 and x = x0 -- h~.,l(O) there exists an open neighborhood U of r / s u c h that U C p r ~ ( B ( x , e)). If x is arbitrary one chooses r = d(xo, x) + e. For v ~ S~ X = 7r-I (x) and e > 0 consider the open neighborhood
C~;(v) = {c,,,(~) I w 6 S ~ X a n d / ( v , w) < e}. Fix a compact set K such that U • v ( K ) -- X and a reference point x() E K. Then the following uniform version of Step 2 holds. STEP 3. There exist R > 0 a n d ~ > 0 such that f o r all p ~ K and x E X
C~:(v) C p r x ( B ( p , R ) ) f o r some v ~ S~-()X. Suppose Step 3 is false. Then there are sequences p , 6 K, x, ~ X such that
Ci/,,(v) Cpr,,,(B(p,,,n)) for all v ~ S~)X. Since x, c Int B(p,,. n) implies pr~,, ( B ( p , , n)) = X (cx~), we can assume after choosing a subsequence that x,, ~ ~ and p,, ~ p ~ K. Then Step 2 implies the existence of r > 0, e > 0 and vr ~ S,~)X such that C~:(v()) C p r E ( B ( p , r ) ) . The continuity of the projection implies the existence of nr such that for all n ~> nr we have: C~./2(v()) C prx,, ( B ( p , , r)). But this contradicts the choice of the sequence if l / n < e / 2 and n > r. The next step follows from the positivity of #/, on open sets.
STEP 4. For all ~ > 0 there, exists a constant g. -- f ( e ) > 0 such that
u,(c
(v)) > e
f o r all v E S,.oX and p ~ K. m
Now consider x 6 X and p E X. Choose y 6 F such that y p ~ K. Since
# p ( P r x ( B ( P , P))) -- # • 2 1 5
P))
the estimate ( l ) follows from Steps 3 and 4. Choose p ~> R and a reference point x() 6 X. Then we call
8,,.'
- prx,,
p))
71
G. Knieper
520
a ball of radius r = e -d(x~ about se = prxo(X) at infinity. For A C X(cxz) and m ~> 0 consider the m-dimensional Hausdorff measure
H m ( A ) - - ,-+o lim inf
rjm rj <~e, A c_ j=!
B~ (~j ) j=l
associated to the "ball"-structure introduced above. Then part (b) of Proposition 2.1 can be used to prove that an c~-dimensional B u s e m a n n density is equivalent to the c~-dimensional Hausdorff measure. More precisely, there exists a constant c > 1 such that 1 - lzxo (A) <~ H ~ ( A ) <~ c lzx,, ( A ) . c
Since there exists an 6 ( F ) - d i m e n s i o n a l Busemann density by Patterson's construction,
ot - - 6 ( F ) = h --dimHausd. X(oo). Furthermore, the estimate implies that all B u s e m a n n densities are equivalent. One even can show that the F - a c t i o n is ergodic. This implies that the density is uniquely determined (see [65] for more details).
5.3. Volume growth and growth rate of regular closed geodesics The local estimate of the Busemann density in Proposition 2.1(c) can be used to obtain some asymptotic properties of the geometry (see [65]). THEOREM 3.1. Let (M = X / F , g) be a compact rank 1 manifold and x ~ X. Then there exists a constant a > 1 such that 1 vol S,.(x) -~< <~a a
e hr
for all r > O. PROOF. Choose p ~> 2R, where R is as in Proposition 2.1 and let x I . . . . . x,,, be a maximal p-separating set in S,.(x), i.e., a set of maximal cardinality such that d ( x i , x j ) >/p for i g: j . In particular, [,_J B(xi, p) D_ Sr(x) and B(xi, p/2) 71 B(xj, p/2) = 0 for i g: j . The local property of the B u s e m a n n density implies 1 -hr
-e b
~ #x(Prx(B(xi, fi) 71Sr(x))) ~ b e - / "
for all t3 with 2R ~> ,6 I> R and a constant b > 0. In particular, !11
,.,.
-''r i=1
Hyperbolic dynamics and Riemannian geometry
521
and
-1m e -hr ~ #x b
I UPr.,. !11 t
"tBrxi, p / 2 ) n Sr(x))
t
~/s
i=!
and, hence,
1 eh r <~ m <~ fle hr
/3
for a constant/4 > 1. From the uniformity of the geometry follows 1
for r I> r0, R <~/5 ~< 2R and, therefore,
Ill vol S,.(x) <. Z
vol(B(xi, p) N S,.(x)) <<.fig e h'
i--I
and 1 eh ,. vol S,.(x)>~ ~'" vol(B(xi, p ) N S,.(x))>~ [4---[ i=!
which yields the claim.
I-1
REMARK. We remark that in the case of negative curvature the method of the proof can be used to derive Margulis' result, namely the existence of the limit
lim r-~ ~
w)l S,. (x) = a (x). e h,
The reason is that in this case we can work with balls B(xi, p) of arbitrarily small size. Furthermore, the function a ( x ) can be interpreted as the total mass of lzx. COROLLARY 3.2. If M = X / F is a compact rank 1 manifold, the Poincard series
Z e-s,l(l~,• diverges for s - h f o r all p, q ~ X, i.e., F is of divergence type.
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522
PROOF. Let .T" C X be a fundamental domain
f~. z e-sd(p'Yq) dv~ ycF
-- y ~ fy --
Since the right hand side diverges for hence, for all p, q ~ X.
fo ~ s =
e -sd(p'q) dvol(q) (.9:)
e -sr vol Sr (p) dr. h,
the Poincar6 series diverges for one and, IN
REMARK. In [65] it is shown that all discrete cocompact lattices of Hadamard spaces are of divergence type. In the following we study the growth rate of regular closed geodesics on rank 1 manifolds. In the case of negative curvature all geodesics are regular and one can use symbolic dynamics and the analytic theory of zeta-functions to obtain much better estimates (see [3]). Now let M = X/F be a compact rank 1 manifold. We are interested in counting primitive (prime) non-oriented closed geodesics, i.e., closed geodesics which are not an iterate of another one. The flat strip Theorem 1.5 in Section 2 implies that the regular closed geodesics are uniquely determined in their free homotopy class. In general this is not true for the singular closed geodesics, since typically, as the example of a flat torus shows, they come in an uncountable free homotopic family. Therefore, we select a maximal set 79(M) of primitive and geometrically distinct closed geodesics representing different free homotopy classes. We denote by 79reg(M), respectively 79sing(M) :-- 79(M) \ 79tog(M) the subset of all regular, respectively singular closed geodesics. Furthermore, 79(t), 79,.cg(t), respectively 79~ing(t) is the subset of geodesics in 79(M), 79,-eg(M), respectively 79.~i,lg(M) of length ~< t. Finally, P(t) = card 79(t), P,-eg(t) = card 79,-cg(t) and &ing(t) = card 79.~ing(t) denote their corresponding cardinality. It is easy to see that under the assumption of nonpositive curvature free homotopic closed geodesics have the same length. Therefore, the counting functions above are independent of the selection 79(M). The free homotopy classes are in one-to-one correspondence with the conjugacy classes of F. Therefore, the set 7a(M) is in one-to-one correspondence with the following equivalence classes in F \ {id}. Two elements YI, Y2 ~ F \ {id} are called equivalent if and only if there exists n, in E Z and r/E F such that •
=
Denote by [F] the set of equivalence classes. We call Y ~ F \ {id} primitive if it cannot be written in the form y = ~'", m t> 2 for some 13 ~ F . Each [y] ~ [F] can be represented as [y] = {r/yr rl -! I Y0 ~ F, Y0 primitive, r/~ F, n ~ Z}. For y ~ F' denote by g(y) - - i n f { d ( p ,
YP) I P E X}
Hyperbolic dynamics and Riemannian geometry.
523
the length of y. Since our manifold is compact, there exists p0 9 X such that d(p0, yp0) = g(y) and the geodesic through po, Ypo is an axis of y. The projection onto M is a closed geodesic of length (prime period) g([y]) -- min{g(r/) I r / 9 [y]} = g(Yo), where Y0 is a primitive element representing [y]. Therefore,
P(t) -- card{[y]
9
IF] I e([•
~< t}.
LEMMA 3.3. There is a constant al > 1 such that P(t) <<.al e hr. PROOF. Fix a fundamental domain .T. Then each [y] 9 [F] can be represented by a primitive element Yo with axis passing through 3m. If g(Yo) ~< t then yo(.T') _ Bt+,.(xo) for some constant c > 0. Since Theorem 3.1 implies vol(Bt+,.(xo)) <~eht/~, for/~ > 0, we obtain P(t) <~/7)eht/vol(f') = a e ht . I-7 In Section 5.6 we will improve this trivial upper bound. For that we will use the uniqueness of the measure of maximal entropy derived in Section 5.5. In this section we study the regular closed geodesics. First we will obtain a better upper bound for P,.c~(t). To do this, we are following an approach which is often used in trace formulas. Define for p, q 9 X
a(l,,q,t) = card{y
9
F ld(p, y q ) <~ t}.
If Xl~.tl: IR ~ I~ is the characteristic function of the interval [0, t ], we can write
a(p, q, t) = Z XI'"tI(d(P' Yq))" yel"
LEMMA 3.4. Let .T be a fundamental domain of'X/N then
f
a(q, q, t)dvol(q) ~< vol B(p, t + D),
where D = diam f and p 9 f . PROOF. Choose p e .T. Since d(p, vq) <~d(p, q) + d(q, yq), we have that a(q, q, t) <~ a (p, q, t + D). Hence, /,
f F a ( q , q, t) dvol(q) <~]~ a(p, q, t + D) dvoi(q)
=
E XI~ ycl"
Yq))dvol(q)
G. Knieper
524
-- • -
LEMMA 3.5.
-
Xl~
q))dv~
fx Xl~
q ) ) d v o l ( q ) = vol
Br+D(p).
D
There is a constant b > 0 such that
O(t)= Z
g.(c) <~be h'.
cc P,.~g(t) PROOF. The lemma above, together with Theorem 3.1 implies the existence of a constant b l > 0 such that bl 9ehr ~> •
= where Conj(M) Conj(y) be the F/Zy, the map representing the
f.,. XlO,~l (d(x, yx)) dvol(x)
E
E
orEConj(M) rE~,
denotes the set conjugacy classes of F. Let Z~, be the centralizer and conjugacy class of y E /-'. If R• C /-' is a set representing the cosets R• -+ Conj(y) with ~1 ~ 71-1Yr/is a bijection. Choose a subset H C / - ' conjugacy classes Conj(M). Then we obtain
h,. eht E E fT x.,,.,,(,(x.,,-I y 71x)) dvol(x) y E I7 tIER ),
=zz/,, y E H tIER),
Xl~).rl(d(x, yx))dvol(x) ([f')
XlO.rl(d(x, y x ) ) yEFI
where f ( Z • is a Z• -- {yl')' In 6 Z} follows since each would bound a flat
dvol(x),
(Zy )
fundamental domain of the centralizer Z• of Y. If Y is regular, then is the infinite cyclic group generated by a primitive element Y0. This element/4 of Z• fixes the regular axis a• of V, since otherwise a• strip. Furthermore, a fundamental domain of Z• is of the form
exp,,yo(,) N~yo(,)--. f • O~
U
where
N,,n~(.,) is the
normal space to the axis a•
U• (5)- Ix E .Y• I d(x, a• <~~}
in a•
Let
Hyperbolic dynamics and Riemannian geometry be the 8-tubular neighborhood of a•
g(Yo)] in f'•
525
For each x E U•
d(x, yx) <~d(x,xo) + d(xo, yxo) + d(yxo, yx) <~2~ + g.(y), where we choose x0 6 a• f(Y0)] such that d(xo, x) <<.8. Denote b y / 7 o the subset of primitive elements i n / 7 and Ho(t) the elements in H() of length smaller than t. Then
bl
>~
XiO,tl(d(x, yxl)dvol(x)
>~
E
~
y E F / o ( T - I)
=
Xlo,tl(d(x, yx))dvol(x) y(l/2)
E
vol U•
yEH()(t- I ) By comparison with Euclidean geometry, we obtain a constant b2 > 0 such that vol U •
~> f ( y ) . b2.
Note, that the set of primitive elements corresponds to the regular closed geodesics 7~,.eg( M ). Hence,
E
b3eht>~
f(Y)--
yEH~(t-
I)
E
f(~')
~'E'P,.c,., (t - I)
DI
for a new constant b3 > 0 which yields the lemma. PROPOSITION
3.6.
There exists a constant a, > 0 such that a)
P,-c,g(t) <~ - e hI. t
PROOF. Define O(t) -- ~,.E.p,~Vtt ) f(C). Let li > 0 be smaller than the length of the shortest closed geodesic. Then
Prc,,(t) .
' d0(s)
ft~ .
.
.
S
O(t) + f' O(s) e hI ~ ds <~ b - - + h t
S-
fl~ t eh'
t
( e ht ) ds=O -7- " K]
Now we are going to obtain a lower bound for the number of regular closed geodesics. Define Frog = {y 6 F I the axis corresponding to y is regular}.
526
G. Knieper
If y 6/"reg then obviously [y ] C Freg. Therefore, if [ Freg] C [/"] are the equivalence classes consisting of regular elements, then Preg(t) -- card{[y] 6 [Freg] I s
~< t}.
LEMMA 3.7. Let s be the length o f the shortest closed geodesic in M, B r ( p ) a ball o f radius r -- s / 4 about p ~ X and c'II~ --+ X be an axis o f the primitive element Yo ~ F. Then card{y IV =/3Y~/3-1 , e(y) ~< t, /3(c) fq B r ( p ) # O] ~< t/r. PROOF. If g(Y0) -- to and y -- r -I with e(y) ~< t then Ikl ~< t/to. If k is fixed, k ~ 0,/~ ?'~/3-I # / ~ y ~ / ~ - I implies /~-I/3 r Z• where Z• is the centralizer of yo. In particular, /4(c) r fi(c) since otherwise /~-~/~ -- y~ for g 6 Z. If/4(c) and /~(c) have nontrivial intersection with B r ( p ) we will find q, ~ 6 c([0, to]) and n , m E Z such that /3y0" (q), r, ~ ,,, implies d ( q , ~) >~.3/2 since #Yo,,, (q) 6 B,.(p). Then g --/3y 0,, -~ ~ --/~V0 .3/2 ~> d ( g ( q ) , ~, (~)) >1 d(~, -I gq, q) - d(~, q) >~s - d(~, q). Hence, for fixed k there are not more than 2to/s different elements of the form y --/-J yCf/~-I such that/4(c) fq B,.(p) ~ 0 and the lemma follows. [--1 Essentially, the same estimate holds if we replace B,.(p) by any compact domain.
COROLLARY 3.8. If K C X is a compact set and c'II~ ~ element )/~ ~ F. Then there exists a constant e such that card{y I v --/JY(~/4-~, g(Y) <<.t, [4(r
X an axis o f the primitive
B,.(p) ~ O} <~ et.
In order to obtain a lower bound for P,-eg(t), we will construct many regular elements in F such that the corresponding axis are passing through a compact domain. For that we need some preparations. For p E X and 0 ~< r < t define ~ " ( p ) = {y ~ F I r < d ( p , Y ( P ) ) <~ t} and ~ ( p ) = ~()(p). Then, for fixed r/> 0, Theorem 3.1 implies card C"(P) ~> ~eh' for a constant b > 0. Moreover, we need the following two lemmata. Consider the metric dl on S M defined in Section 5.1 and denote by B I (v, r) the ball of radius r about v E S M with respect to di. LEMMA 3.9. Let c be a hyperbolic: axis q f rl ~ neighborhoods U o f c ( - o o ) , V o f c ( + o o ) and all Y ~ I-" with d(xo, yxo) <<.t and Y U f-) V -- 0 endpoints in U and V. Moreover, h(O) ~ B I (~'(0),
I-" and x o - c(O). For e > 0 there are constants p > O, n ~ 1~ such that f o r the axis a o f ~"Y~" is hyperbolic with e) and f(O"yrl") <~ t + p.
Hyperbolic dynamics and Riemannian geometry
527
LEMMA 3.10. Let c be a regular axis o f rl 9 F. I f xo = c(O), there are constants r, k > 0 such that card{y 9 l-'t+r(XO) l y U n V--91} ~ l/4cardFtk(xo). The proofs of Lemmata 3.9 and 3. l 0 can be found in [62]. THEOREM 3.1 I. There are constants a3 > 0 a n d to > 0 such that Prcg(t) ~> a3. e h t / t
forallt
> to.
PROOF. If M is a compact rank 1 manifold the regular closed geodesics are dense. Choose a regular axis c : R ---> X and define for e > 0: A~:(t) "-- {V 9 F I f ( y ) ~< t the axis a of ), is regular, h(0) 9 B I (~(0), e)}.
It follows from Lemmata 3.9 and 3. l0 that card A~.(t + q) >~ 1/4card Ft k (x0), where q = p + r. Corollary 3.8 yields card{[ylOA~(t)J~ 0 and a3 > 0 such that
Preg(t) >/
card A ~ ( t ) eh I >/a3 /t et
for all t ~> to.
[-I
REMARK. Since the regular closed geodesics are dense, the above argument shows that the number of regular closed geodesics passing through any open set grows exponentially. More precisely, if O C S M is any open subset in S M and p..eg(t )o _ card{c Ic closed, primitive, and regular, g(c) ~< t, ~' n O -r ~4} then Pr~~ ( t ) >1 b e t ' ' / t
for a constant D > 0 depending on O.
G. Knieper
528
5.4. Construction of an ergodic measure of maximal entropy Now we construct, using the Patterson-Sullivan measure # p , an invariant measure for the geodesic flow (see [66]). Let G be the space of nonparametrized geodesics in X and
be the set of points in X (cxz) • X (e~) which can be connected by at least one geodesic. According to its definition, # p is F-quasi-invariant with R a d o n - N i k o d y m cocycle
f ( Y , ~) --e-hi'P(•
p'~). For (se, r/) ~ g7E consider
flp(~, rl) = -(bp(q, ~) + bp(q, rl)), where q is a point on a geodesic c connecting ~ and r/. In geometrical terms fl/,(~, 17) is the length of the segment c which is cut out by the horoballs through (p, ~) and (p, 7/). Since gradq bp(q, ~) = - g r a d q bp(q, rl) for all points on geodesics connecting ~ and rl, this number is independent of the choice of q. An easy computation shows: LEMMA 4.1. For p ~ X d/~(~, r/) = e/'~'1~''1) d#/,(~) d#/,(r/)
defines a F-invariant measure on G/" Let P : S X --+ ~/': be the projection given by P(v) = ( c , , ( - ~ ) , c,,(cx~)), where c,, is the geodesic with ~',,(0) = v. Note that the projection 7 r : S X ~ X maps the fibers of P onto geodesics or flat totally geodesic submanifolds of X. Therefore, the flow acts isometrically on the fibers of P. Then ~ induces a 4~t-invariant measure # on S X by setting for each Borel subset A C S X #(A)-
f~7, vol(Tr (P - t (8, '1)A A))d/](~, 0),
where vol is the induced volume element on the submanifolds. By the F-invariance of this measure descends to a flow-invariant measure on S M. We normalize it so that it becomes a probability measure, i.e., # (SM) = I. In order to compute the entropy of the measure # we consider a measurable partition .,4 = {A I . . . . . Am} of S M such that the diameters of all elements in .,4 are less than e with a / ' ) , then, by the definition respect to the metric dl introduced in Section 5.1 9 If v 6 a 6 ~,,/, of .,4, it follows that It--I = c
A k=()
8,,, (e
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529
LEMMA 4.2. Let 0 < e < min{R, inj(M)}, where inj(M) is the injectivity radius o f M.
Then there is a constant a > 0 such that #(or) <~ e - h " a f o r all ot E A(n) ~4~ 9 PROOF. Since cr C j('-3nu,=0I ~-k Bj~ (q~kv, e) if v E cr we have for all w E ot that d ( c v ( t ) , c,,,(t)) ~< e for all t E [0, n]. Let p 6 X be the reference point used in the definition of the measure/_z and fi E S X be a lift of v such that d(rrfi, p) ~< diam M = r0. Since e < i n j ( M ) we can lift the set c~ to a set 6~ e S X such that for all tb e (~ we have d(c,~,(t), c~,(t)) <<.e for all t E [0, n]. Let c~(n) = x and ~ = c,~,(-o<~). Let C~.x be the geodesic connecting and x such that c~,,. (n) = x. Then,
d(c~,x(O), p) <~ d(c~,x(O), 7r(ffo)) + d(yr(ffv), p) <~ 2e + r(,, i.e., ~ E p r x ( B ( p , r() + 2e)). Therefore, if P : S X ~ we have
P(a) C
U
{'l}
•
~!~: denotes the endpoint projection,
pr,,(B(x,e,)).
qCpt'~ ( B(p.r()+2~'))
For each 71E p r x ( B ( p , r() + 2e)) choose a point q E B ( p , r() + 2~) that lies on the geodesic c,~.,. Then, using the transformation rule for the Busemann density, Proposition 2.1 and the estimate d ( q , x) ~> d ( x , 7r~) - d(zr ~, p) - d ( p , q) >>.n - r(i - (2e + r0), we obtain
l~t,(pr,,(B(x, e))) <~ eh'l(t"q)lzq(pr,,(B(x, e))) ~ eh('"'+e~;)be-h"(q") <~ [~e -h'' for a constant/~ > 0. Since l z v ( p r x ( B ( p , ro + 2e))) ~< ~ l , ( X ( o o ) ) < cxz the proof follows from the definition of it. Fl THEOREM 4.3. The measure lz is a measure r h~ ( ~ ) = h ( ~ )
entropy f o r the geodesic flow, i.e.,
=h.
PROOF. Choose a partition A as above. Then by L e m m a 4.2
H(A~'))= ~ /z(c~)(-log/z(c~))>/(hn-loga)Z /~(ot)--hn-loga. EA(") c r . , "4,
~
EA(") " "4,
Hence, h ( ~ , A) ~> h. The theorem follows since h >~ h1,(dp) >~ h ( ~ , A) >~ h.
F-l
530
G. Knieper
Now we show the ergodicity of/z. For v ~ S X and ~ = cv(oo) let W" ( v ) -
{w 6 S X I w -- - g r a d b ~ r v ( q , ~ ) , b ~ r v ( q , ~ ) - - 0 }
denote the strong stable manifold associated to v. PROPOSITION 4.4. Let v E S X be a recurrent regular vector. Then f o r all w E W s (v)
a~ (0' (~), , ' (~)) --, o as t --+ cx~.
PROOF. Since v is recurrent there exist sequences y,, E F and t,, --~ ~ with Dv,,(ckt"(v)) ~
v
as n ~ ~ .
If w is asymptotic to v the function t ~ dl (4~~(v), 4~~(w)) is m o n o t o n e decreasing. If it does not converge to 0, there exists a constant bl such that dl (4) t (v), 4~t (w)) ~> bl for all t ~> 0, and we conclude b, ~< d,
(~,,,+, (v), 4,,,, +, (u,)) <~ d l ( v , u ,)
for s E l - t , , , cx~). Therefore, b, <~ d l ( 4 9 ' D y , 4 f l " ( v ) , 49' D y , r
<~ d , ( v , u,).
Passing to a subsequence if necessary we can assume that Dy,, (ckt"(w)) ~ then we obtain
w' E S X . But
h, <~ d~ (r (v), r (to')) <~ d, (v, w) for all s 6 I1~. Hence, 4r' (v) bounds a flat strip by the flat strip theorem contradicting the fact that v is regular. [-1 Let a be a rank l axis. By L e m m a 1.4 there exist open neighborhoods U, V C X (cxz) of a ( - o o ) , a ( + ~ ) such that each ~ E U and r/E V can be connected by a rank 1 geodesic c which, therefore, is unique up to parametrization (see [6] for details). Define G,-,~c(U, V) - {c geodesic I i" recurrent, c(-cx~) E U, c(+cx~) E V }. This set has full measure in the set of all geodesics with endpoints in U and V with respect to /z. Let f : S X ~ R be a continuous F-equivariant function. Then, by the Birkhoff ergodic theorem,
f + (c) "-- "r~o~limTI ~ 7" f ( ~ ( t ) ) d t
Hyperbolic dynamics and Riemannian geometry
531
exists and is equal to
l f, 7. f ( 6 ( - t ) ) d t
f-(c)'=
lim -T~ T
for/x-almost all geodesics c. Note that f + and f of c. Consider the set ~rcc(U, V ) = [c 6 ~rcc(U, V) l f + ( c ) -
are independent of the parametrization
f-(c)}
which has full measure in {7,-c,:(U, V). By the structure of the measure # there exists a geodesic cl 6 {Tree(U, V) with cl ( - c o ) -- ~ such that G~ -- {r/6 V
I c ( + ~ ) = o, c(-oo) - ~; and f + (c) - f - ( c ) } c V
has full measure with respect to/zp. LEMMA 4.5. f + is constant a.e. on G,.cc(U, V). PROOF. For almost every c ~ O,-c~.(U, V) we have c ( + ~ ) 6 G~. Let c2 be the geodesic such that c 2 ( + ~ ) -- c(+cx~) and c2(-cxz) -- ~ -- t'l ( - ~ ) . Then Proposition 4.4 implies: lim d,(~'2(t),~'(t))--
lim
d,(~'2(t).~',(t))--O.
The continuity of f implies that f+(c)
-
THEOREM 4.6.
f+(c2)
-
f-
(c2) -
f-
(el) -
D
f+(cl).
The measure lz is an ergodic measure with re,wect to the geodesic flow
on SM. ,-,.,
PROOF. Let f ' S X ~ IR be a continuous F-invariant function and V -- {71 6 V 171 = c.(+eo), c 6 {7,.co(u, v)}. It follows from L e m m a 4.5 and Proposition 4.4 that f + is constant a.e. on the set of geodesics c with c ( + o o ) E V. Consider the set
yEP
We know that Y A V has full measure in V with respect to ~;, and, hence, with respect to #q for any q. Every ~ E X(c~) has an open neighborhood V(~) that can be moved into V by an element of F . Therefore, Y n V (~) has full measure in V (~) for each ~ 6 X (co) and, hence, Y has full measure in X ( ~ ) . Consequently, the set Z = {c I c ( + ~ ) 6 Y} has full measure with respect to #. Then, by the F-invariance of f + , f + is constant a.e. on Z. Hence, the measure/z is ergodic with respect to the geodesic flow. I-1
532
G. Knieper
COROLLARY 4.7. The regular set in S M has full measure with respect to #.
PROOF. The discussion above shows that the regular set has positive measure. Since the regular set is flow-invariant, the corollary is a consequence of the ergodicity of/z. [--1 In the next section we sketch a proof of the uniqueness of the measure of maximal entropy, obtained in [66].
5.5. The uniqueness o f the measure o f maximal entropy Following Bowen [12] we call a h o m e o m o r p h i s m T: V ~ V on a compact metric space (V, d) entropy expansive if its local topological entropy vanishes. More precisely, for x E V and e > 0 let Z~:(x) = {y E V I d ( T " x , T " y ) <~ e for all n E 1~1}is the e-neighborhood of the orbit {T"X},E's If h ( T , Z~:(x)) = 0 for all x E V we call T entropy expansive with expansivity constant e. Under this assumption, for any T-invariant probability measure v, h,,(T) = h,,(T, A ) provided A = {Ai . . . . . Am } is a Borel partition of diameter less than e. Furthermore, it is also easy to see that the map v w-~ h,,(T) on the set of T-invariant Borel probability measure M ( T ) is upper semi-continuous. It is a simple consequence of the convexity of the distance function that the geodesic flow on a compact manifold M of nonpositive curvature is entropy expansive. More precisely, one has: PROPOSITION 5.1. For each k E I~ one has that the time k-map ckk o f the geodesic flow on S M is entropy e~pansive. Choosing the metric dk on S M the number ~ = i n j ( M ) / 3 provides an expansivity constant f o r qbk.
Before we can sketch the proof of the uniqueness of the measure of maximal entropy we need to fix some notations. Let R be the constant introduced in Proposition 2.1 and p ~ .U be a fixed reference point in a fundamental domain f . If A C X we define S A to be the set of unit vectors with footpoint in A. For x E X and R > 0 let
D ( x , R' , R) -- {v E S B ( p ,
R
f
)lc,,(r)EB(x
,
R) f o r s o m e r > 0 ]
.
STEP 1. Let R' > 2R + 1/2 be a constant sati.~l~.,ing ~ C B ( p , R') and d ( p , X) >~ R' + R. Then we obtain
p ( O ( x , R' , R)) >~ c ~e -tul(p'x) f o r a constant c' > 0 not depending on X. This is a consequence of the local property of the measure # p . The next step follows from the convexity of the distance function. STEP 2. The cardinality o f (d,, 6)-separated set o f D ( x , R', R) is bounded from above by a constant a = a(6, R', R) f o r all x E X such that d ( x , p) ~ n + R' + R.
Hyperbolic dynamics and Riemannian geometry
533
Now consider a maximal 2R-separated set xl . . . . . xk(,,) of the geodesic sphere S ( p , r(n)) of radius r(n) = 2n + R' + R about p. Since this set is 2R-separated, the balls are pairwise disjoint and since it is maximal, the balls cover S B ( p , R'). Thus we can choose a partition F~" . . . . . F k(n) 2'' of S B ( p R') such that
D ( x i , R', R) C Fi21' C D ( x i , R', 2R) for each i. Let Q : S B ( p , R') --+ S M be the restriction to S B ( p , R') of the projection ~n S X --~ S M and let L 2n i - - Q ( F i- ) . Note that the map Q is at mostc~ to one for some constant u > 0. S T E P 3. L2n -- {,., 112n . . . . .
L2nk(,)} is a covering such that each v E S M lies in at most ot
sets from L 2'' 9 Furthermore, there is a constant [4 > 0 such that # ( L 2I" ) >~ f i e -2h'' f o r all i E {1 . . . . . k(n)}. The first assertion follows since Q is at most c~ to one. The second assertion follows from Step 1 and the fact that ~ ( A ) ~< o t ~ ( Q ( A ) ) for all measurable sets A C S B ( p , R'). To prove the uniqueness of the measure of maximal entropy we have to deal with partitions of small diameter. Let V = {vl . . . . . v,,,} be a maximal (d2,, ~)-separated set of S M . Choose a partition B such that for each B C / 3 there exists vi E V such that
B,12,, ( vi, ~/2) C B C B,i,,, (vi, ~). STEP 4. card{BE/31BNL
i211 =/=0]<~a(e.
, R t
,2R)
,
where a (s, R', 2R) the constant is the one defined in Step 2. Since
211
L i
--
Q( F i
3-.11
) we can lilt
to an e-separated set of F i2" C D ( x , R , 2R). Therefl)re, the result follows from Step 2. Now consider the pushforward A"
2n -- {qb"L i
} of the covering L2" by the time n map of
the geodesic flow. It follows from the definition of L 2'' that for each A E A" and any two vectors v, w E A there exists a continuous curve c~ :[0, 1 ] --~ S M connecting v and w such that the length of rr o 05tc~ is bounded by R' for all t E [--n, n]. The reason is that the set Fi2'' is contained in D ( x i , R ' , 2R). We will use the covering A" to separate measures in the following way. STEP 5. Let v, ~ be mutually singular Borel probability measures on S M such that the singular set has measure 0 with respect to I~. Then there exists a union C, o f subsets o f A" such that #(C,,) ~ O, v(C,,) ~ I f o r n ~ cx~.
534
G. Knieper
PROOF. Since # s v there exists a m e a s u r a b l e set I2 s.t. #(S2) = 0 and v(a'-2) = 1. By assumption we can assume that the singular set is contained in S-2. For each 3 > 0 one can choose c o m p a c t sets KI C a"-2 and K2 C SM\I-2 such that v ( K i ) ~> 1 - 3 and l z ( S M \ K 2 ) < 3. N o w define
Cm " - - U { A
6A'" IANKI
#dp }.
Then there exists n 6 1N such that for all v 6 K i, w 6 K2 and lifts fi, tb 6 S X d(rr4/fi, rrq~' tb) ~> R'
for some t 6 [ - n , n].
Otherwise one could prove, using a c o m p a c t n e s s argument and the flat strip T h e o r e m 1.5, the existence of a vector wo 6 K2 such that the geodesic c,,, 0 bounds a flat strip. This would contradict the fact that K2 is contained in the regular set. The set C,z covers K i by definition and C,, N K2 = ~b by the choice of n and the definition of A". Consequently, v(C,,) >~ 1 - 6 a n d / z ( C , , ) ~< l z ( S M \ K 2 ) <. 3. IS] The next standard estimate is used in the proof.
-- Z
ai l~ ai <~
i=1
ai
l
logk+-
i=1
(5.1)
e
provided a l . . . . . ak ~> 0 and Y~Y=I ai <~ 1. In the last step we will show that the measure of maximal entropy is unique. STEP 6. Let v be a Borel probability measure invariant under the geodesic flow. If v # then ht.(q5) < h = h,(49). PROOF. S i n c e / z is ergodic it is enough to prove that h~,(4~) < h if v and l~ are mutually singular. Fix e ~< i n j ( M ) / 3 and let V = {vl . . . . . v,,, } E S M be a maximal (e, d,, )-separated set. Choose a partition/3" such that for each B E/3" we have B,/~,, (vi, e / 2 ) C B C B,12,, (vi, ~) for some vi ~ V. Proposition 5.1 implies
2nh,,(ck') - h,,(ck2") = h , , ( c k 2 " , l Y ) <~ H,,(13") - - -
Z
v(B)logv(B).
B 613 ,t 1 2 n . . . . . . 1 2kn } of S M constructed between Step 2 and Step 3 Consider the covering L 2'' -- {'-'1
and its push forward {~b"L~"} = A". Since v and # are mutually singular and the singular set has measure 0 with respect to/~, Step 5 implies the existence of sets C,z consisting of a union of elements of A" such that # ( C , , ) ---, 0 and v(C,,) --+ 1 as n --~ cx~. Using the above estimate (5.1), we obtain: 2nh,(4~) ~< -
Z {1:IG13n Id)n Bfq(_'n#~}
v(B) log v(B) { B~13"
y~ 14~"BnC,, =r
l)(B) log v(B)
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G. Knieper
5.6. Growth rate o f singular closed geodesics In this last section we come back to the various counting functions of closed geodesics. Keeping the notation of Section 5.3 we denote by 79(M) a maximal set of primitive and geometrically distinct closed geodesics representing different free homotopy classes and by 7")reg(M) respectively 79sing(M) :-- 79(M) \ 7")reg(M) the subset of all regular respectively singular closed geodesics. Moreover, P(t), Preg(t) and esing(t) is the number of geodesics in 79(M), ~')reg(M) respectively ~sing(M) of length ~< t. It is a consequence of the uniqueness of the maximal measure, which we will denote by /Zmax that the topological entropy h(q~lsing ) of the geodesic flow restricted to the singular set is strictly smaller than the topological entropy h of the unrestricted flow. This can be seen by the following argument. The variational principle (see [57] or [83]) implies the existence of a sequence/z,, of invariant probability measures concentrated on the singular set such that
h(~lsing)/> hi,,, (4)) ~> h(qblsing) It is not difficult to prove that the map v --+ h,,(4~) defined on the space of 4~-invariant probability measures is upper semicontinuous with respect the weak topology [66]. Let/z be a weak limit of l~,,. Since the singular set is a closed subset of S M the limit measure is concentrated on the singular set as well. Therefore, the upper semi-continuity and the variational principle imply ht~ (4~) = h(q~lsing). On the other hand the maximal measure is concentrated on the regular set. Therefore, ~ # ~m,,x and the uniqueness of the measure of maximal entropy yields h > hj~ (4~) = h(q~lsing). As we will show, this implies that there are exponentially less singular than regular closed geodesics. THEOREM 6.1. There erists an e > 0 and t~) ~ 0 such that f o r all t >~ to P~ing(t) -~;t <~e Prcg(t) PROOF. Let a = lim sup I ---~,"x~
log P,;i ng (t) it
be the exponential growth rate of the singular closed geodesics. Since closed geodesics corresponding to different free homotopy classes are g-separated for some 6 > 0 it follows from the definition of the topological entropy that a ~< h(q~lsing). Hence, the theorem follows for any e > 0 smaller than h - h(q~lsing). I--I REMARK. For surfaces a stronger result holds. All orbits in the singular set have zero Lyapunov exponents and therefore the topological entropy of the singular set is zero. With the same argument as above one deduces that the growth rate of singular closed geodesics is subexponential. However, in higher dimensions graph manifolds provide rank 1 examples with an exponential growth rate of singular closed geodesics [66].
Hyperbolic dynamics and Riemannian geometry
537
As a last application, we obtain an estimate for the number of all closed geodesics. THEOREM 6.2. There exists a constant a > 1 such that f o r all t > 0 1 eh I < < . P ( t ) < ~a- e ht at t
PROOF. In Section 5.3 we derived such an estimate for the regular closed geodesics. Therefore, the lower bound is obvious and the upper bound follows from Theorem 6.1. [-1
6. Appendix 6.1. An intrinsic p r o o f o f Pestov's identity In this appendix we will give an intrinsic proof of a certain formula of Weitzenb6ck type on the tangent bundle of a Riemannian manifold. This formula, discovered by Pestov and Sharafutdinov [81], is called "Pestov's identity". The main difference to their formula is that it is stated in an invariant fashion avoiding the use of local coordinates. This identity is of particular interest for manifolds of nonpositive curvature and it should have many applications in dynamics and rigidity. For instance it has b,'en used by Croke and Sharafutdinov to show that compact manifolds of negative are is~,spectrically rigid, generalizing previous work of Guillemin and Kazhdan, and Min Oo (see Section 4.4). We will use the following notations and definitions. As usual let T M be the tangent bundle, S M the unit tangent bundle of a Riemannian manifold M, and 7r the canonical projection. Let ~0 6 C ~ ( T M ) be a infinitely often differentiable function and denote by grad~0(v) -- (grad h ~p(v), gradV q)(v)) 6 T , , T M the gradient of 99 with respect to the Sasaki metric (see Section I.I). The horizontal and vertical components grad h q)(v), grad v ~o(v) 6 T~rI,,IM are uniquely determined by the identities (grad" ~o(v), u,)"--
d 1=()
and {grad v 99(v), u,)"--
d
99(v + tu,) 1_--()
where for v, u, c T p M t --+ v,,,(t) denotes the parallel translation of v along the geodesic c,1, with initial conditions ~',,,(0) -u,. The horizontal and vertical gradient are examples of semibasic vector fields, i.e., smooth
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G. Knieper
542
Now observe that (grad v XGqg(v), grad h qg(v)} = {gradV{gradh qg(v), v), grad h qg(v))
--(Vgradh,p(v)grad h qg(v), v} + {grad h qg(v), grad h qg(v)}. Using this together with Step 1 and Step 2, we obtain Pestov's identity.
17
Now we are going to integrate Pestov's identity over the unit tangent bundle with respect to the Liouville measure dttr --d0p d vol(p), where d0p is the measure induced by the Riemannian metric on S p M and d vol is the associated volume element on M. LEMMA 1.2. Let X : T M --~ T M be a semibasic vector field. Then for each p ~ M divhX(v) d 0 p ( v ) = d i v ( L ' /, M
X(v)d0p(v))
/, M
and
,
div v X(v)dO/,(v)
/, M
L"I,M
+
- ,). f
I,M
REMARK. If X is homogeneous of degree m (i.e., X (iv) = i'" X (v) for i > 0), it follows that VVX(v) -- dl~=0X((l + t)v) -- m X ( v ) and consequently f div vX(v) d0p(v) - - ( n + m - l ) /
L, /, M
{X(v), v}dOp(v).
JS/, M
THEOREM 1.3. Let M be a compact n-dimensional Riemannian man~yold and q9 E C ~ ( T M). Then 2 L'M (gradh qg(v), grad v X(;qg(v))dpL
If q9 is homogeneous of degree m, we obtain
2 f'M {gradh ~o(v), grad v XGqg(v))dlzL
Hyperbolic dynamics and Riemannian geometry
- L" Ilgradh~~ M
543
+ (n + 2m) L M
LM
(R(grad v ~o(v),
v)v, grad v qg(v)) d/zL.
Acknowledgement I would like to thank Norbert Peyerimhoff and Karl Friedrich Siburg for for carefully reading this survey and useful discussions.
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[53] J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer-Verlag (1998). [54] J. Jost, Nonpositive Curvature: Geometric and Analytics Aspects, Lectures in Mathematics, ETH Ziirich, Birkhauser (1997). [551 M. Kanai, Geodesic flows on negatively curved manifolds with smooth stable and unstable foliations, Ergodic Theory Dynamical Systems 8 (1988), 215-239. [56] A. Katok, Entropy and closed geodesics, Ergodic Theory Dynamical Systems 2 (1982), 339-367. [57] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Vol. 54, Cambridge University Press (1995). I581 A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math. 98 (1989), 581-597. [59] A. Katok, G. Knieper and H. Weiss, Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows, Comm. Math. Phys. 138 ( 1991 ), 19-31. 1601 W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. 99 (1974), 1-13. [61] W. Klingenberg, Riemannian Geometry, 2nd ed., de Gruyter, Berlin (1995). [62] G. Knieper, Das Wachstum der /iiquivalenzklassen geschlossener Geodiitischer in kompakten Riemannschen Mannig['altigkeiten, Arch. Math. 40 (1983), 559-568. 1631 G. Knieper, Spherical means on compact Riemannian manifolds of negative curvature, Differential Geom. Appl. 4 (1994), 361-390. [641 G. Knieper, Volume growth, entropy and the geodesic stretch, Math. Res. Lett. 2 (1995), 1-20. [65] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, GAFA 7 (1997), 755-782. [66] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 man~fi~lds, Ann. of Math. 148 (1998), 291-314. 1671 E Ledrappier, Harmonic measure and Bowen-Margulis measures, Israel J. Math. 71 (1990), 275-287. 1681 A.N. Livgic, Some homological properties of U-systems, Mat. Zametki 10(1971 ), 555-564. 1691 R. de la Llave, Canonical perturbation theory r Anosov systems and regularity results for the Liv.~i~" cohomology equation, Ann. of Math. 123 (1986), 537-611. 1701 R. Mafi6, On a theotenr of Klingenberg, Dynamical Systems and Bifurcation Theory, M. Camacho, M. Pacilico and F. Takens, eds, Pitman Research Notes in Math., Vol. 160(1987), 319-345. 1711 A. Manning, Topological entropy for geodesic flows, Ann. of Math. 110 (1979), 567-573. 1721 G.A. Margulis, Certain measures associated with U-[tows on compact manifi, hls, Funct. Anal. Appl. 4 ( ! 97()), 55-67. 1731 M. Min-Oo, Spectral rigidity for manifolds with m'gatiw" curvature operator, Contemp. Math. 51 (1986), 99-103. 1741 M. Morse, A fim&tnu'ntal class of geodesics on any ~'losed surfiwe of genus greater than one, Trans. Amer. Math. Soc. 26 (! 924), 25-60. [751 J.-P. Otal, Le .wectte marqm; des surfaces h courbure mXgative, Ann. of Math. (2) 131 ( i 990), ! 51-162. 1761 G.P. Paternain, Geodesic Flows, Birkhfiuser, Boston (1999). 1771 S. Patterson, The limit set of Fuchsian group, Acta Math. 136 (1976), 241-273. 1781 Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surv. 32 (1977), 54-114. 1791 D. Ruelle, An inequality.fi~r tire enttrq~y ofdifferentiable maps, Bol. Soc. Bras. Mat. 9 (1978), 83-87. 180] T. Sakai, Riemannian Geometry, Transl. of Math. Monographs, Amer. Math. Soc. (1992). 1811 V.A. Sharafutdinov, Integral Geometry of Tensor bTelds, VSP, Utrecht, The Netherlands (1994). 1821 D. Sullivan, Tire density at infinity of a discrete group r motions, Inst. Hautes l~tudes Sci. Publ. Math. 50 ( 1979), 171-202. 1831 P. Waiters, An hrtrodm'tion to Ergodic Theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin (1982).
6
Hyperbolic Dynamics and Riemannian Geometry Gerhard Knieper Fakultiit f~r Mathematik, Ruhr- Universitiit Bochum, NA 05/33, 44780 Bochum, Germany E-mail: gknieper@ math. ruhr-uni-bochum, de
Contents I. Basics on Riemannian geometry and geodesic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
455
I.I. Levi-Civita connection and the linearization of the geodesic flow . . . . . . . . . . . . . . . . . . .
455
1.2. Jacobi equation
459
..............................................
1.3. Geodesic Anosov flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Geometry of manifl~lds of nonpositive curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Hadamard manilolds
...........................................
473 476 476
2.2. Rigidity of the sphere at inlinity in negative curvature . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Busemann functions and the horospherical foliations . . . . . . . . . . . . . . . . . . . . . . . . . .
479 485
2.4. Volume growth and entropy formulas
487
..................................
2.5. Poincard series and Buscmann densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Horospherical foliations
............................................
3. i. Ergodic theory of the loliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Regularity of the foliation and rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Entropy and rigidity
..............................................
489 491 491 492 495
4. I. Entropy comparison and rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
495
4.2. Regularity of topological entropy
5(X)
....................................
4.3. Minimal entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
502
4.4. Spectral rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
509
4.5. Rigidity of the marked length spectrum
515
.................................
5. Ergodic properties of weakly hyperbolic spaces 5. I. Definition of rank and rank rigidity
...............................
...................................
515 515
5.2. Uniqueness of the Busemann density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Volume growth and growth rate of regular closed geodesics . . . . . . . . . . . . . . . . . . . . . .
518 520
5.4. Construction of an ergodic measure of maximal entropy . . . . . . . . . . . . . . . . . . . . . . . .
528
5.5. The uniqueness of the measure of maximal entropy
532
5.6. Growth rate of singular closed geodesics 6. Appendix
..........................
................................
....................................................
6. !. An intrinsic proof of Pestov's identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. I A Edited by B. Hasselblatt and A. Katok 9 2002 Elsevier Science B.V. All rights reserved 453
536 537 537 543 543
454
G. Knieper
Abstract In this survey we will focus on recent results on the dynamics of geodesic flows on compact manifolds of nonpositive curvature with hyperbolic or at least weakly hyperbolic behavior. Furthermore, we will discuss rigidity theorems on compact negatively curved manifolds like entropy rigidity, the minimal entropy problem, rigidity of stable and unstable foliations and the infinitesimal spectral rigidity theorem. In many cases we will provide complete proofs and not only statements of theorems. Some of the results are new and did not appear somewhere else. For instance, we give a new upper bound for the number of closed geodesics in rank 1 manifolds and a new entropy comparison result. Moreover, we will give an intrinsic proof of a formula of Weitzenb6ck type on the tangent bundle (Pestov's identity). However, we will not discuss geodesic flows on manifolds with positive curvature. The reason is that until now there is no example of such a flow with a hyperbolic set of positive Liouville measure. Since we do not leave the geometric setting, we do not discuss the interesting subject of general Euler-Lagrange flows. Many of those topics are covered in the recent book by G. Paternain [76] which is in some sense complementary to this survey. Furthermore, we do not cover geodesic metric spaces as for instance Gromov hyperbolic spaces or the Alexandrov geometry of nonpositively curved metric spaces. This theory is treated in many text books and surveys as for instance [6,16,42].
Hyperbolic dynamics and Riemannian geometry.
455
1. Basics on Riemannian geometry and geodesic flows 1.1. Levi-Civita connection a n d the linearization o f the geodesic f l o w As a general reference for the basic theory on Riemannian geometry we refer to the following sources [21,53,61,80]. Let M be a smooth (usually C ~ ) manifold, denote by T M ~ I , ~ M T p M its tangent bundle and let n" 9T M ~ M be the canonical projection. A family g - - ( , ) -- {gp}p~M
of inner products gp on T p M depending differentiably on p is called a Riemannian metric. The tangent space 7',, T M of T M has for every v 6 T M a natural linear subspace tangent to the fibers which is, therefore, generated by smooth curves X : ( - s , + e ) --+ T M , X (0) = v such that ~ l , = 0 7 r X ( s ) - 0. This space is called the vertical space V,, and is given by I/',, = ker dTr,,. The Riemannian metric g induces a natural horizontal space H,, via the Levi-Civita connection D. This connection is uniquely determined by the fact that it is torsion free and that the metric g is parallel. The horizontal subspace H,, of T,, T M is generated by parallel /) vector fields, i.e., by smooth curves X" ( - s , + s ) --~ T M , X(O) -- v such that ~ X ( s ) - - 0 , where ~17 denotes the covariant derivative along the curve y (s) "-- Jr o X (s). Hence, if we define the connection map C,, : 7",,T M --~ Tn,,M to be
C,, ~
.s=()
X(s)
-~,,.:o
'
the horizontal space H,, is equal to the kernel of C,,. Since H,, O V , , - {0} and dim H,, = dim V , , - d i m M , we obtain T~,TM -- H,, 9 V,,. Therefore, the map (drr,,, C,,)" T , , T M ---> T~,,M • T~,,M given by ~ ~ (dn',,(~), C,,(~)) is a linear isomorphism. Consequently, H,,~{(u',())Iu'ETrr,,M}
and
V , , ~ { ( 0 , u ' ) I u , ETzr,,M}.
The geodesics c in M are smooth curves with parallel velocity vector, i.e., /7i. -- 0. They are also characterized by their energy minimizing property. The Riemannian metric induces a distance function d on M. In the sequel we will assume that the metric space (M, d) is complete. By the theorem of H o p f - R i n o w this is equivalent to the fact that M is geodesically complete. Denote by c,, :F~ ---> M the geodesic with initial condition (',,(0) = v. Then the geodesic flow 4)t : T M ---> T M is given by 4)t (v) = b,,(t). Therefore, d dt
4)' (v) ~ t =()
(dl ~
Jrr (v), 0
)
- (v, 0),
t =()
i.e., the infinitesimal generator of the geodesic flow is X ( ; ( v ) -- (v, 0). The main reason for introducing the above decomposition of T T M in horizontal and vertical space is that it
G. Knieper
456
provides a convenient way for computing the differential of the geodesic flow using Jacobi fields. Jacobi fields are solutions of the linear equation
J"(t) + R ( J ( t ) , ckt (v))ckt (v) = 0 , where
J: =
(1.1)
D j and R(X, Y ) Z denotes the curvature tensor.
LEMMA 1.1. The differential
D e t (v)'Trr,,M x Tjr,,M--+ Tjrr
x Trrr
is given by DckI (v)(x, y) = (J(t), J'(t)), where J(t) is the Jacobi field along cv with J (0) -- x, J:(O) -- y. PROOF. Let X ' ( - e , X (0) = v
+e) --+ T M be a smooth curve such that and
D
d ds
X ( s ) = y.
Jr X ( s ) = x , s =0
s =0
Then, using the chain rule and the definition of the horizontal and vertical spaces, we obtain ?) D e I (v)(x, y)"--
,=or
8 o X (s)
D
Jr o r
-
~ ,=or ~X (s)).
s =()
Since (s,t) -+ rr o 4; o X(s) defines a geodesic variation of the geodesic 7r o r the horizontal component J ( t ) " - - ~il [s=()7r o t~t X(s) is a Jacobi field and it's covariant derivative D]
D 0
8
~r x(s) - ~
J ' ( t ) = t-~ ~-~s s----{)
~r
x (,,,) -
D
s =()
r s =()
l-I
is the vertical component. There is a natural l-form (,9 on T M defined by CO,,(~) = (v, dn,, (~)). Using the decomposition of ~ introduced above, we obtain
(x, y ) ~
T,,TM in the horizontal and vertical part,
fO,,(x, y) -- (v, x). LEMMA 1.2. The differential d69 is a symplectic form given by dO,,((xl, Yl), (x2, Y2)) - ( y l , x2) - (y2, xl).
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G. Knieper
458
Hence, the infinitesimal Hamiltonian vector field submanifolds of constant We will mainly study manifold of energy 1/2) SM-
{v E Y g
generator X G ( V ) = (v, 0) of the geodesic flow is the of the kinetic energy. In particular, the flow leaves the energy invariant. the geodesic flow on the unit tangent bundle (the sub-
l llvll - 1},
where Ilvll- (v, v)!/2. The tangent space T v S M is given by {(x, y) l x , y E T~rr
y 2_ v},
since for each smooth curve X ' ( - e ,
+ e ) ---> S M with X(0) -- v it follows that
0 - ~1,-0(x(s), x ( s ) ) - (~l,=0x(s), v). (b) Denote by ECb(v) = s p a n { X , ; ( v ) } -- {(Xv, O) l~. ~ R} the 1-dimensional space tangent to the geodesic flow at v E S M , and by E 4' the associated vector bundle. The interesting part of the dynamics occurs transversal to the flow lines. The next lemma shows that there is a canonical invariant bundle transversal to the flow. LEMMA 1.4. The orthogonal complement
N ( v ) "-- E ~ ( v ) Ain T~,SM, with respect to the Sasaki metric, defines a bundle N invariant under the linearization o f the geodesic flow. Furthermore, N is a symplectic subbundle, i.e., the symplectic f o r m restricted to N is nondegenerate. PROOF. Since N ( v ) -- {(x,y) I x 2_ v, y 2_ v} the nondegeneracy follows from the definition of w; the invariance follows from L e m m a 1.3. [-1 Assume that dim M = n -+- l and, therefore, the dimension of the bundle N is equal to 2n. In order to describe the linearization of the geodesic flow restricted to N it is convenient to trivialize N along the flow lines 4)t (v) in the following way. Choose a parallel orthonormal frame El (t) . . . . . E , ( t ) orthogonal to 4~I (v) and consider for t 6 R the linear map p, " R" • R" ----~ N(dp' (v)) given by
p t ( x , y) - -
yi Ei(t)
x i Ei(t), i=1
,
Hyperbolic dynamics and Riemannian geometry
459
where X i and yi are the coordinates of x and y. Denote by w0 and go the standard symplectic and Euclidean structure on R" • R n, i.e., 090((Xl,Yl), (x2, Y 2 ) ) - - y l
" X2 - - X l " y 2
and
go((xl, yl), (x2, y2)) -- xl "X2 -k- Yl " Y2, where x 9 y -- Y]'i'= ! x i y i is the standard scalar product on II{". For each t ~ R the map p, transforms the symplectic structure oJ and the Sasaki metric gs restricted to N ( r (v)) to the standard symplectic and Euclidean structure on IR" x IR". Furthermore, the linearization D e ' (V)lN(v):N(t~) ~ N ( r t (19)) can be described in the following way. The curvature tensor induces for each t a symmetric endomorphism R ( t ) : r (v) • --~ r (v) • given by R ( t ) w = R(w, r (v))r (v). Associated to the parallel orthonormal frame, we obtain the symmetric matrix representation R(t)i.i = ( R ( E i ( t ) , Ct(v))r (v), E j ( t ) ) which we also denote by R(t). With respect to this framing, the Jacobi equation (1.1) translates into the second order ODE on R" (1.2) which we call Jacobi equation as well. Now, Lemma l.l immediately implies. LEMMA 1.5. If J(t) -- ( j l ( t ) . . . . . J " ( t ) ) is a solution of the Jacobi equation
)'(t) + R ( t ) J ( t ) --0,
(I.2)
then (J(t), j ( t ) ) -- p~-i
De' (v)p{}(J(O),
)(0)).
In the next section, we will study the Jacobi equation in the framework of linear symplectic geometry on a Euclidean vector space. For the basic terminology in symplectic geometry we refer to [26].
1.2. Jacobi equation Let (E, ( , ) ) be a vector space with scalar product ( , ) . Let End(E) denote the set of endomorphisms and let Sym(E) denote the set of symmetric endomorphisms on E. If R" It{ -+ Sym(E) is a smooth C ~ - m a p (curvature operator), the linear differential equation ~176
J ( t ) -k- R ( t ) J
-- 0
is called the Jacobi equation and its solutions J" ~ ~ B" ]R --+ End(E) is a solution of the matrix equation
B(t) + R ( t ) B ( t ) - - 0 ,
E are called Jacobi fields. If
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Hyperbolic dynamics and Riemannian geometry
461
REMARK. The Wronskian and the symplectic structure are related in the following way. If A, B :[a, b] ---> E n d ( E ) are C ! -curves, then
(F2(A(,), B(t))x, y ) - co((A(t)x, .4(t)x), (B(t)y, /}(,)y)) for all x, y e E. LEMMA 2.2. Let X, Y : R ~ E n d ( E ) be Jacobi tensors. Then
F2(X(t), Y(t)) = const for all t e IR. PROOF. Since X ( t ) x and Y(t)y are Jacobi fields, the proof follows from the remark above. [--I Important notions in symplectic geometry which are also useful in our context are isotropic and Lagrangian subspaces. DEFINITION 2.3. A linear subspace L C E x E is called isotropic, iff w(~e, 71) : 0 fl)r all ~, ~1e L. L is called Lagrangian, iff L is isotropic and dim L = dim E.
REMARK. Since c,) is nondegenerate, the dimension of isotropic spaces is always less than or equal to dim E. LEMMA 2.4. Let A, B : E ~
E be endomorphisms o r E and
L -- {(Ax, B x ) l x e E I .
Then L is isotreq~ic if and only i['A r B = B ~A. If additionally x ~ (Ax, Bx) is in.jective, L is Lagrangian. In particuhtr, the graph L - - { ( x , Ax)[ x e E I
is l_ztgnmgian ~['and only i['A is symnletrir PROOF. The proof follows from o)((Ax, Bx), (Ay, By)) -- (Bx, Ay) - (Ax, By) --(A r Bx, y ) - (B' Ax, y).
I---1
REMARK. Note that L = {(Ax, Bx) Ix r E} is a graph if and only if A is nonsingular. DEFINITION 2.5. Let R :Ii~ ~
S y m ( E ) be a smooth map. A family of Lagrangians Lt C E x E, t E [a, b], with Lr = 7,J'I.,,L, is called a Lagrangian deformation of L,, induced by R.
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Hyperbolic dynamics and Riemannian geometry
Integration yields
A(t)-- B(t) fa I (BTB)-I(s)BT(a)ds. This implies that A is nonsingular and, therefore, there are no conjugate points.
C1
The variational aspects of the Jacobi equation is, as we will see, revealed by the index form. DEFINITION 2.8. Let R : R ---> S y m ( E ) be a curvature operator and V = {X : [a, b] ---> E I X piecewise differentiable} be the vector space of continuous and piecewise differentiable curves in E. The symmetric bilinear form I := ll.,t,l: E x E - + R
with
f
II,,,,,I(X, Y ) =
h
(X(t), ~ ' ( t ) ) -
(R(t)X, Y)dt
is called the index form. If we want to specify the curvature operator, we write I k or I k
I.,/'1"
REMARK. Let X, Y 6 V be differentiable on each interval of the subdivision a = to < tt < 9.- < tk -- b. Then, using integration by parts, we obtain
k I(X, Y)-- Z ( , Y , Y) i--I
Ii li t I
c Ii'(J( + R(t)X, V)dt. J,
(2.2)
This implies that X is a Jacobi field if and only if I(X, Y) = 0 for all Y :[a, b] --+ E with Y(a) = Y(b) = 0. Since
I(X + s Y , X + s Y ) = 21(X, Y),
s=() X is a Jacobi field if and only if it is a critical point of the action I(X) : = I(X, X) on the space of vector fields with fixed end points. This is not surprising, since the Jacobi equation is nothing else than the E u l e r - L a g r a n g e equation for the time dependent Lagrangian 1
L(t,x, 3')- -~((Y, Y) - (R(t)x,x)) on[a,b]•
E x E.
464
G. Knieper
LEMMA 2.9 (Index L e m m a ) . Assume that the Jacobi equation has no conjugate points on [a, b]. Then I[a,hl is positive definite on the subspace
V~
{X 9 V I X ( a ) -
X(b)--O},
i.e.,
I(X,X)>0
for all X 9 V ~ X r O.
PROOF. Let us first assume that X 9 V ~ is smooth. Consider the Jacobi tensor A with initial conditions A(a) = 0 and ,4(a) = id. Since the Jacobi equation has no conjugate points, there is a smooth curve Y 9 V such that X(t) = A(t)Y(t). In particular, Y(b) = 0 . Then we obtain I(X, X) -- (X, X)l~i~ -
fh
(~( + R(t)X, X)dt
1'(~4"Y + R(t)AY, AY)dt ---
(2A~+
AY A Y ) d t
2(A );', A Y ) + ( ~ ' , A / A Y ) d t
--2
(~./irar)dt +
(~, (arar))dt.
Since L e m m a 2.2 implies
O _ A / f~ _ f~r A, we obtain
( A r A y ) -- f~l+A y + A r f~ Y + A/+A ) / _ 2 f~/A y + AT+A)/. Hence,
I(X, X) --
(f/,ArAf/)dt.
In the general case, we deduce the same formula by integrating piecewise. This shows I(X, X) ~> 0 for X 9 V (). Furthermore, I(X, X ) - - 0 implies that Y is constant. Since Y(b) = 0, we deduce that 0 = Y(t) = X(t) for all t 9 [a,b]. [7 REMARK. In particular, this l e m m a shows for Jacobi equations without conjugate points on [a, b] that there are no conjugate points inside the interval [a, b] as well. To see this, one considers for a ~ t l < t2 ~< b the piecewise differentiable curve X which is 0 on [a, t i]
Hyperbolic" dynamics and Riemannian geometry
465
and [t2, b] and coincides with a Jacobi field J on [tl, t2] with J (tl) -- J ( t 2 ) - - 0. Then it follows from the formula (2.2) that I(X, X) -- 0 and the Index L e m m a 2.9 implies that X vanishes identically. The next corollary shows that for Jacobi equations without conjugate points, Jacobi fields are not only critical points but minimizers of the action I(X, X) on the space of piecewise differentiable curves with fixed end points. COROLLARY 2.10 (Minimizing property of Jacobi fields). A s s u m e that the Jacobi equation J ( t ) + R ( t ) J ( t ) = 0 has no conjugate points on [a,b]. Let J be a Jacobi field and X ~ V be a piecewise differentiable field with X (a) = J (a) and X (b) -- J (b). Then ~
l(J, J ) <~ I(X, X), where the inequality is strict unless J ~ X.
PROOF. From the Index Lemma 2.9 and the formula (2.2), we deduce 0 ~< l ( J - X, J - X) -- I(J, J ) - 2I(J, X) + I(X, X) -- - l ( J , J ) + I(X, X). 71
This inequality is strict unless J -- X.
In the sequel, we like to compare Jacobi fields associated to a pair of different curvature operators. For that we will need the following partial ordering on S y m ( E ) , where for A, B 6 S y m ( E ) we define A ~< B iff ( A x , x ) <~ ( B x , x ) for all x E E. In the standard Rauch comparison estimates, one of the occurring curvature operators is a constant multiple of the identity. Therefore, the following notation is useful in Rauch's comparison estimates. For k ~ It~ denote by sk(t) 6 C ~ ( I ~ ) the solution of the differential equation ?(t) + k s ( t ) - 0
with initial conditions s(0) - 0 and .~'(0) -- 1, i.e.,
sk(t) --
/
~ - k sinh v/-L-kt
for k < 0,
t ~i- sin x/kt
fork = 0 , for k > 0.
PROPOSITION 2.11 (Rauch). Let R'It~ --+ S y m ( E ) be a C ~ - m a p and J a Jacobi field with J (0) -- 0 and [[J (0)[[ = I. (a) A s s u m e that R ( t ) <<,kl id a n d r <~ 7r/v/kl i f k l > O. Then [[J(t)l[/skl(t) is monotone increasing on [0, r] and b o u n d e d below by !. I f f o r some to > O, we have [[J(t0)[[- skt (t()), then J (t) - sk, ( t ) . J (0) f o r all t 9 [0, t0].
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G. Knieper
468
by applying the above estimate to the solution C ( t ) of the Jacobi equation ~149
C ( t ) 4- R ( r - t ) C ( t ) - - 0
with C(t) -- B ( r - t). Applying the Index L e m m a 2.9 to the curve J(s)-
I A(s)A-I(t)x' I B(s)B-l(t)x,
O<~s<<.t, t<~s<~r,
we obtain 0 < Ii0,,.l(J, J) --(,4(t)A -I (t)x, x ) - (/~(t)B -I (t)x, x) and, hence, the lower bound.
D
LEMMA 2.13. For T >~ 1 and [4 > 0 let R" I~ ~ S y m ( E ) be a curvature operator without conjugate points on [ - 1 , T + 1] and -/4 2 id ~< R(t). Then there exists a constant p > 0 depending only on [3 such that f o r all Jacobi fields J ' [ - 1 , T + 1] --~ E with initial conditions J ( O ) - - 0 and I l J ( O ) l l - 1
IIJ (r)ll >1 p f o r all r e [ 1, TI. PROOF. Fix r e [ 1, T] and consider the piecewise differentiable vector field X ' [ - 1 , 1] ~ E with
X(t) -
0, J(t), (l-(t-r))J(r),
r 4-
- 1 ~
Then O
)-
f
r+l
(R(t)X(t),X(t))dt.
According to Corollary 2.12, we obtain for the Jacobi tensor A with initial conditions A (0) - - 0 and A (0) -- id, the estimate ][/~(r)[[ <~ ][,4(r)A-I(r)l[]lA(r)[] ~ fl[[A(r)[[cothfi. Consequently, IlJ(r)ll ~/~'llJ(r)ll, where fl'--/4 coth ft. This yields for the index form I - - Ii_l,r§ I 0 < I(X, X) ~ IIJ(r)l12(3 ' -4- 1) 4- 3 2
f
r+l
(X(t), X(t))dt
I..a
E]
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0
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G. Knieper
470
LEMMA 2.14. Let R" I~ --+ S y m ( E ) be a curvature operator without conjugate points on [ - 1, oo) and Sr the Jacobi tensor with Sr (0) = id and Sr (r) = O. Then the limit
S(t)-- lim Sr(t) r---->oo
exists and S(t) is nonsingular f o r all t >~O.
PROOF. Since I2(S~,, Sr) = 0, we have that Sr(0) e Sym(E). Furthermore, r ~ Sr(0) is strictly monotone increasing, i.e., (S,. (0)x, x) < (Se (0)x, x) for all x e V, x -r 0 and 0 <~ r < g. Namely, if
X(t)=
Sr(t)x 0
for 0 ~< t ~< r, forr~
and J (t) = Se (t)x, we obtain, using the index form, Ii(}"el and the minimizing properties of Jacobi fields (2.10) -IIo.tI(X, X) --(S,. (O)x, x) < -Ii{},,~l(J, J) --(Se (O)x, x). On the other hand, S,. (0) is bounded above by S-I (0). To prove this, consider
J ( t ) --
S_l(t)x S,.(t)x
for -- 1 ~< t ~< 0, for0 ~< t <~ r.
Using the Index Lemma, we get 0 < ll_,.,.l(J, J) --(S_, (0)x, x ) - (S,.(())x, x). Hence, the limit lim,.~ ~ S,. (0) --" S(0) exists which implies the assertion by the continuous dependence on initial conditions. S(t) is nonsingular for t >/0, since otherwise there are r > 0 and x e E with S ( r ) x = 0. Because there are no conjugate points we would have S,.(t)x -- S ( t ) x for all t ~> 0. This would contradict the fact that (S,.(0)x, x) is strictly monotone increasing. I--1 DEFINITION 2.15. If R" ]t~ ~ S y m ( E ) has no conjugate points we call S(t) = lim S,.(t)
the stable Jacobi tensor, and U(t) =
lim
S_,.(t)
F ----~ Or
the unstable Jacobi tensor.
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G. Knieper
472
for all t ~> O. Since lim r---~ ~
sinh c~(r + s) = e us sinh c~r
for a l l s 6 I ~ ,
the inequalities in (2.4) follow. Since S ( - t ) is the unstable solution of , 4 ( t ) + R ( - t ) A (t) = 0, we obtain (2.5) from (2.4). If - f 2 i d ~< R(t) Rauch's comparison estimate implies that I I A ( s ) x l l / s i n h ~ s is monotone decreasing, and we obtain for 0 ~< t ~< r and x 6 E \ {0} IIA(r - t)xll /> sinh/3(r - t)
II A (r)x II
sinh/3r
and IlA(r + t)xll ~< sinh/3(r + t)
IIA (r)x II
sinh f r
Hence, as in the first case, we obtain the estimates in (2.6) and (2.7).
l-q
LEMMA 2.17. Assume that -/4 2 id ~< R(t) and that R has no conjugate points. Then IIS(t)" S-I (t)ll ~ -/~ coth~r. Similarly, we obtain for U,.(t) "= S_,.(t) that /),. (0) <~/~ coth fir. As in the proof of Lemma 2.14 one shows using the Index form I1_,. r I that
o<
x ) - (s
Hence -/-Jcoth/Jr ~< S,.(0) ~< /),-(0) ~< /3coth/4r and, therefore, -/3 ~< S(0) ~< /)(0) ~< /~. F--I Consider the symplectic maps ~Pt,()" E x E ---> E x E with ~t(z) = (J:.(t), J : ( t ) ) , where J is the Jacobi field with (Jr. (0), Jr (0)) -- z. If the Jacobi equation has no conjugate points one defines the stable and unstable spaces E" and E" by
E'={(x,S(0)x)lxeE} and E"--{(x,U(0)x)lx~E}.
473
Hyperbolic dynamics and Riemannian geometry
The following proposition shows the hyperbolicity of the Jacobi equation in the case of a negative definite curvature operator. In particular, this proposition implies that the geodesic flow on a manifold of negative curvature is an Anosov flow (see Section 3). PROPOSITION 2.18. Assume that there are constants 0 < ot < fl such that _f12 id ~< R(t) ~< --0~2 id.
Then E s ~D E u = E x E and there are constants 0 < B <~ A such that ne-t~'llzll ~ II~,(z)ll ~ A e-'*rllzll
for z 9 E", and B e c'' Ilzll ~< II~P,(z)ll ~< A e t~r Ilzll
for z 9 E", where Ilzll- v/llxll e + Ilyll 2 for z -- (x, y) 9 E x E. PROOF. If Z = (x, S(0)x) 6 E ~, we obtain (J:(t), J: (t)) -- (S(t)x, S(t)x) and
]l(J:(t)' J-(t))ll 2 - IIJ:(t)llr "
-+- ( llJ-(t)ll ii~(t)ll )
-IIS(t)xll
v/ (,, 1+
IIS(t).,cll
Since by Lemma 2.17 IIS(t)xll
IIS(t)xll we obtain from Lemma 2.16 e -t~' Ilxll <~ IIJ:(t)ll
% II(J:('), J:('))II <
ll:llV/ +
Since I1=11= v/llxll2-t - IIS(0)xll 2 ~ Ilxllv/1 +/./2 the first estimate follows. The second estimate can be derived in the same way. gl
1.3. Geodesic Anosov flows DEFINITION 3.1. Let ( M , g ) be a (n + l)-dimensional complete Riemannian manifold and II" II the norm on T S M induced by the Sasaki metric. The geodesic flow 49r:SM -~ SM is called Anosov flow if there exist constants k, C > 0 and a splitting
T~,SM = E' (v) 9 E" (v) 9 E~(v)
G. Knieper
474 such that E4~(v) = span{XG(v)} and
IIDr (v)~ II c
e -k' I1~ II
for all ~ 9 E ' (v), t ~> 0, as well as
IID r
(v)~" II ~
c e -k' ][~ 1[
for all ~ 9 E" (v), t ~> 0. REMARK. (a) If M is compact this definition does not depend on the choice of the norm. (b) The definition and the fact that Dgt acts isometrically on the bundle E ~ imply that ES(v) and E"(v) are subspaces of (ECb(v)) -L = N(v). The flow invariance of the symplectic structure co and the inequality Iw(~, r#)[ ~< I1~11" Ilqll imply that E" and E" are isotropic subbundles of N and thus, their dimensions are not bigger than n. On the other hand, E" (v) ~ E" (v) = N(v), and, therefore, the subbundles are Lagrangian. Furthermore, the definition implies that they are invariant and continuous. (c) The definition also implies that I ek t
IIDqb-'(v)~ II ?
li il
for t >~ 0 and ~ 9 E' (v), and [IDdp1(v)~ll
?I
ek t
ii li
for t ) 0 and ~ 9 E"(v). PROPOSITION 3.2. Let ( M, g) be a complete Riemannian manifold with -[42 <~ K <~ -or 2 for constants [3 ~ ot > O. Then its geodesic flow is Anosov. PROOF. Fix an orbit 4t (v) and consider the trivialization Pt :R" x R" ---> N(ck~(v)) of the normal bundle N along 4~~(v), associated to a parallel orthonormal frame Ei (t) . . . . . E , ( t ) orthogonal to 4/(v) (see Section 1 and, in particular, Lemma 1.5). Then Dck~(v)= p~p~p-~i, where lp~ (z) = (Jr.(t), j:(t)) and Jr is the solution of the Jacobi equation f - ( t ) + R(t)Jr.(t) with (J:.(0), Jr.(0)) -- z and Rii(t) -- (R(Ei(t) ~v)dptv, Ei(t)). The curvature assumption implies -/4" id ~< R(t) ~ - o r - i d and from 2.18 we obtain a decomposition of IR" x IR'z into subspaces E" and E" such that B e -t~' Ilzll ~ II~,(z)ll ~ A e -u' Ilzll for z E E", and B e~
~ II~,(z)ll ~ A e~'llzll
Hyperbolic dynamics and Riemannian geometry
475
for z e E". Since Pt is an isometry, the proposition follows for E" (v) -- pt]-I E" and EU (v) = p(-~l E u. 0 The following theorem is due to Klingenberg [60] and shows that only a restricted class of compact manifolds can carry geodesic Anosov flows. THEOREM 3.3 (Klingenberg). Let (M, g) be a compact Riemannian manifold such that the geodesic flow is Anosov. Then there are no conjugate points. REMARK. Together with the Theorem of Hadamard and Cartan (see Section 2, Theorem 1.1) it follows that the universal covering of compact manifolds carrying an Anosov geodesic flow has to be diffeomorphic to IR". It is an amazing result due to Marl6 that the conclusion of Theorem 3.3 holds under a much more general assumption [70]. THEOREM 3.4 (Marl6). Let (M, g) be a complete Riemannian manifold o f finite volume. Assume that the bundle N -- ( ECb)• normal to the geodesic flow qbI admits a continuous Lagrangian subbundle invariant under the geodesic flow. Then (M, g) has no conjugate points. IDEA OF THE PROOF. The existence of a Lagrangian bundle can be used to associate to any closed continuous curve c: I ~ S M an index which counts the crossing of the Lagrangian bundle with the vertical bundle V,, = {(0, w) l w _k v} (Maslov index) [26]. The index does not change under homotopic deformations of c. Using this fact, Marl6 proved that the intersection of the Lagrangian bundle with the vertical bundle is trivial, i.e., the bundle is a Lagrangian graph. Hence, in view of Lemma 2.7 there are no conjugate points (see [70] for more details). [-1
In the same paper, Marl6 claims a generalization of Klingenberg's theorem to noncompact manifolds with geodesic Anosov flows provided the sectional curvature is bounded from below. However, Burns noticed a mistake in Mafi6's arguments [17]. Since certain statements remain valid and might be useful for future research in this area, we will briefly sketch his ideas. Denote by C ( S M ) = {v e S M I c,, :[0, oo) -+ M has no conjugate points} the set of vectors in the unit tangent bundle corresponding to geodesics with no conjugate points. It is not difficult to prove that for all Riemannian manifolds C ( S M ) is a closed subset of S M . Moreover, this set is nonempty if M is not compact. Under the assumptions that the geodesic is Anosov, Marl6 claimed that this set is open as well. The proof is based on Proposition II.2 of his paper which, unfortunately, is not correct as stated there. However, the following weaker version is true. It is a consequence of the Jacobi field estimate given in Lemma 2.13. LEMMA 3.5. Let (M, g) be a Riemannian manifold with lower sectional curvature bound. If the geodesic flow is Anosov there exists a constant cr with the following property. If E" (v) A V (v) r {0} then the geodesic cv has conjugate points on the interval [ - 1, o'].
G. Knieper
476
PROOF. Suppose the geodesic c v ' [ - 1 , T + 1] -+ M, T ~> 1, has no conjugate points. Consider 0 # ~ = (0, ~2) E E s (v) fq V (v) such that 11~2[I = 1. If J is the Jacobi field with J (0) = 0, J ' ( 0 ) = ~2 then, according to L e m m a 2.13 and the definition of E s, p ~< IlJ(t)ll ~< V/IIJ(')II 2 + IIJ'(t)ll 2 - - I I D ~ ' ( ~
II ~
Ce-k'
for all t e [ 1, T]. Therefore, T ~< (log C - log p ) / k -- ?' and the geodesic cv" [ - 1, o-] --+ M has conjugate points provided o" > Y + 1. 17 REMARK. Marie's mistake consists in the claim that, under the assumption of L e m m a 3.5, the geodesic c,, should even have conjugate points on [0, cr ]. From that it is easy to deduce that a geodesic c~, : [0, cxz) --> IR has no conjugate points provided it has no conjugate points on [0, or]. This would have shown that the set C ( S M ) is open. OPEN PROBLEMS.
(a) Is it true that also noncompact manifolds with geodesic Anosov flow do not have conjugate points? (b) It follows from the discussion in [60] that the universal cover of a Riemannian manifold with geodesic Anosov flow is hyperbolic in the sense of Gromov [42]. Is it true that each compact manifold with a geodesic Anosov flow carries a metric of negative curvature'?
2. Geometry of manifolds of nonpositive curvature 2.1. Hadamard man(folds We begin with the following theorem which is foundational in the theory of Hadamard manifolds (see, e.g., [21,53,39,61,80]). THEOREM 1.1 (Hadamard-Cartan). Let ( M , g ) be a complete n-dimensional Riemannian man~fold with no conjugate points. Then, for all p E M, the exponential map expp " TI, M -+ M is a covering map. In particular, the universal covering of M is diffeomorphic to IR". REMARK.
(a) The topology of those manifolds is to a large extend determined by the fundamental group since the contractibility of the universal cover implies that the higher homotopy groups are vanishing, i.e., n'k (M) -- 0 for k ~> 2. (b) If M i, M2 are compact manifolds with nonpositive curvature such that 7rl (M l) and n'l (M2) are isomorphic, then, by a result of Farell and Jones, Mi and M2 are homeomorphic. However, in general they are not diffeomorphic [29,30]. DEFINITION 1.2. A Hadamard manifold is a simply connected complete Riemannian manifold of nonpositive curvature.
Hyperbolic dynamics and Riemannian geometry
477
REMARK. According to Theorem 1.1, Hadamard manifolds are diffeomorphic to R". Furthermore, there is a unique geodesic connecting any pair of points. The most important single tool in the study of Hadamard manifolds are convexity properties of the distance function. A function f : X ~ IR is called convex if for all geodesics c : R ~ X the composition f o c :JR ~ I~ is convex in the usual sense. A set B C X is called convex if for any pair of points p, q 6 B the connecting geodesic is contained in B. In particular, if f : X --+ IR is convex the set {x E X I f ( x ) ~< oe} is convex for all c~ ~ ]R. The following are examples of convex functions. Let d : X x X ~ IR be the distance function on a Hadamard manifold X induced by the Riemannian metric. (a) If cl, c2 : N ~ X are geodesics then t ~ d(cl (t), c2(t)) is convex. (b) If B C X is convex then the function x ~ d ( x , B) is convex. (c) Busemann functions are convex (see Section 2.3). PROOF. The statements in (b) and (c) are consequences of (a), whereas (a) follows from the second variation formula of arc length. I-7 Denote by
,.)
(b) f~ ~> f i + f;_ -- 2glf2cosc~3. (c) g3 ~< g2 cosot'l + gl cosc~2. In each case equality holds if and only if the sides o f the triangle span a totally geodesic flat triangle. COROLLARY 1.4. Let Pt, P2, P3 and p4 be distinct points and consider the quadrilateral with angles Oil --
As a consequence one obtains the so-called flat strip theorem. THEOREM 1.5. Let ct, c2" IR ~ X be two geometrically distinct geodesics. If d(cl (t), c2(t)) is bounded by a constant independent o f t the geodesics span a flat totally geodesic embedded strip. PROOF. Since t --> d(cl (t), c2(t)) is convex and bounded it has to be constant. Then one obtains from the second variation of arc length that for all tl < t2 the points cl (tl), c2(tl ), cl (t2), c2(t2) span a quadrilateral of angle sum 2rr. Using 1.4, one obtains the result. 1--1
G. Knieper
478
The asymptotic geometry of Hadamard manifolds is intimately related to the long time behavior of the geodesic flow and plays a central role in rigidity theory. We begin by introducing the following fundamental notions. DEFINITION 1.6. Let ( X , d ) be a metric space and A C X. For r ~> 0 the r-tubular neighborhood of A is the set
Tr(A) := {q E X I d(q, A) <~ r}. If A, B C X are subsets of X,
dH(A, B ) = inf{r I A C T,.B, B C TrA} denotes the Hausdorff distance of A and B. REMARK. It is easy to see that dH fulfills the triangle inequality. DEFINITION 1.7. Let X be a Hadamard manifold. We call two geodesics cl : R X, c2 : R ~ X asymptotic if cl.(,'.(R+).c2(R+))
< ~.
DEFINITION 1.8. The set of equivalence classes of asymptotic geodesics is called the sphere at infinity and denoted by Xgc,,d(cx~), or simply X ( ~ ) , if there is no confusion of the meaning. For p e X consider the map .ft, :SI, X -+ X(cx~) such that v --+ [c,,], where c,, is the geodesic with b,,(0) = v and, where [c,,] denotes the equivalence class of c,, :II~+ ~ X in X(cx~). This map is a bijection. The topology on X ( ~ ) , defined such that fl, becomes a homeomorphism, is called the sphere topology. Since fi/ o ] ) , l ' S i , X ~ SqX is a homeomorphism this topology is independent of p. The topologies on X(cxz) and X have a natural extension to X := X t2 X (c~) and are characterized by the fact that the map
~: B~ ( p ) - {v ~ T~,M I Ilvll ~ 1} ~ X with 99(v)--
{ (' t expp
fp(v),
I-Ilvtlv ,
[Ivii < 1,
Ilvll- 1,
is a homeomorphism. This topology has been introduced by Eberlein and O'Neill [28] and is called the cone topology. In particular, X is homeomorphic to a closed ball in ~ " . The relative topology on X(oo), respectively, on X coincides with the sphere topology, respectively, the topology on the manifold X.
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2.2. Rigidity o f the sphere at infinity in negative curvature For Hadamard manifolds of strictly negative curvature the sphere at infinity has amazing rigidity properties first discovered by Morse and further investigated by Gromov culminating in the work of Gromov hyperbolic metric spaces [42]. In these spaces it is possible to describe X(oo) using quasi-geodesics which are objects much more diffuse than geodesics. Their advantage, however, is that they are much more robust than geodesics. For example, a quasi-geodesic for a given metric will still be a quasi-geodesic for all biLipschitz-equivalent metrics. Let us first recall the notion of a quasi-isometry. DEFINITION 2.1. Let ( X i , d l ) and (X2, d2) be two metric spaces. A map F" (XI, dl) -+ (X2, d2) is called a quasi-isometric embedding if and only if there exist constants A > 1, c~ > 0 such that 1
--dl (x, y) - c~ ~< d2(Fx, F y ) <~ Adl (x, y) + c~. A A quasi-isometric embedding F is called a quasi-isometry if it has a quasi-isometric inverse G" X2 ~ X! such that
d,(GoF(x),x)<~p
and
d2(FoG(y),y)~
for a constant p > 0 and all x E X i and y 6 X2. In this case, X I and X2 are called quasiisometric. A quasi-isometric embedding q9" (R, l I) ~ (X, d) is called a quasi-geodesic. REMARK. If we want to specify the constants we call such maps (A, c~)-quasi-isometric embeddings ( ( A, c~)-quasi-isometries, ( A, c~)-quasi-geodesics). Let X be a Hadamard manifold. We extend the equivalence relation of being asymptotic to the much larger class of quasi-geodesics. Denote by the equivalence classes of quasi-geodesics, and by Xgc,,d(oo) the equivalence geodesics. Denote by [ ]gc,,d, [ ]quasi the corresponding equivalence classes. The following theorem implies that we do not loose information if we geodesics to quasi-geodesics.
geodesics Xqua~i(oo) classes of pass from
THEOREM 2.2. Let (X, g) be a Hadamard manifold such that K e <, - k 2 < O. Then the inclusion Q " Xgc,,d(oo) ~ Xquasi(oo) with [C]geod ~
[C]quasi is bijective.
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It is obvious that Q is a well defined injective map. The nontrivial part is to show that to each quasi-geodesic ~0 there is a geodesic representative c, i.e., dH(~(R+), c(R+)) < oo. We will prove a somewhat stronger result which is a modern version of a lemma by Morse (Morse lemma) [74]. THEOREM 2.3 (Morse). Let X be a H a d a m a r d manifold o f negative sectional curvature K <. - k 2 < O. Let qg:~, ~ (X, dg) be a (A, ot)-quasi-geodesic. Then there exists, up to parametrization, exactly one geodesic c such that dH(c(It~), qg(II~)) <~ ro, where ro depends only on A, ot and k.
The proof will be a consequence of the following considerations. Let c : R --+ X be a geodesic and P,. : X --+ c ( R ) be the orthogonal projection, i.e., P,. is the identity on c('~) and for p r c(R) the projection P,,(p) is the unique element of c(/R) such that the geodesic segment connecting p and P,.(p) is perpendicular to c(R). By the convexity of geodesic balls, P,. is well-defined and has the following properties. LEMMA 2.4. Let X be a Hadamard manifold o f negative sectional curvature K <. - k 2 < O. Then we have: (a) For all p e X d(p,c(R))=d(p,P,.(p)).
(b) P,. is differentiable and IIDP,.(p)vll <~
c o s h k d ( p , P,.(p))
Ilvll
f o r all p e X and v e T I, X.
(c) For p > 0 there exists y = y ( p , k) > 0 such that f o r all p e X we have diam P,.(B(p, p)) <~ y e -k'I(p'pIpl). PROOF. Statement (a) follows from the convexity of geodesic spheres and (b) follows from the upper curvature bound together with a generalization of Rauch's comparison theorem (see 2.11). Statement (c) is a consequence of (b). D LEMMA 2.5. Let tp : • --+ X b e a n (A, ~)-quasi-geodesic and ot' = u + there exists xl . . . . . x,, e qg([a, b]) such that d ( x i , x i + l ) ~ 2Or' and qg([a, b]) C 0
1. Then, f o r a < b,
B(xi, 2ct'),
i=1
where n = [A(b - a)/ot'] and [r] denotes f o r r e R the largest integer smaller than r.
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P R O O F . L e t a < tl < . . . < tn <~ b be the a ' / A equidistant subdivision of [ a , b]. Since ~o is a (A, c~)-quasi-geodesic, we have for all s E [a, b]
d (qp(s), (p(ti)) ~ or' q-ot < 2~' (.gt
"' where ti = i -X + a is provided we choose 1 <. i <~ [ A ( b - a ) / ~ ' ] such that Is - ti[ <~ -X, an element of the subdivision introduced above. Consequently, xi = cp(ti) are the required elements of ~0([a, b]). I--] LEMMA 2.6. Let ~o : lR --+ X be an (A, c~)-quasi-geodesic a n d c : IR --+ X be a geodesic. Then there exists a constant ro(A, or, k) with the f o l l o w i n g property: For all a < b a n d r >~ ro(A, ~, k) such that cp([a, b]) C X \ TrC(]R) and
d(~o(a),c(IR)) <. r
+ oe
and d(~o(b),c(IR)) <<.r
+
~,
we have: diam(P,.~o(la, bl)) ~< 1.
PROOF. Lemma 2.5 implies the existence of xl . . . . . . ~c,, E ~o([a, b]) such that
cp([a, b]) C 0
B(xi, 2c~'),
i=1
where oei --oe + 1 and n <~ ~A( b - a). Using Lemma 2.4 and the lower bound in the definition of an (A, c~)-quasi-geodesic, we obtain
i=1
A2 A Ib - a l y e -k' <<"7 (d(~o(a) ' Io(b)) + o~)y e - k ' <~ ny e -k'' <~ -~7 The assumption, together with the triangle inequality, implies d(~o(a), ~o(b)) <~ 2r + 2or + I.
Consequently, there exists a constant r0(A, a~, k) such that A2 A2 g ~< -~-7(2r + 3a~)V e - k ' + otye?' e - k ' <~ 1/2 + e/2 for all r ~> r()(A, or, k).
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Hyperbolic dynamics and Riemannian geometry
483
PROOF OF THEOREM 2.3. Let qg" R -+ X be an (A, ct)-quasi-geodesic and let t, ~ I~ be a sequence such that t,, -+ cx~ as n --> cx~. L e m m a 2.7 implies that for any sequence of geodesics c, "]K --> X with c , ( a , ) -- 99(-t,,) and c ( b , ) -- r the estimate dH(qg([--t,,, t,,]), c([a,,, b,,])) ~< rl
holds. Choosing a parametrization of the geodesics such that
c,, ( 0 ) - Pc,
(~0(0))EBr, (~0(0)),
we conclude that a, -+ - e ~ and b, ---->+oo. Furthermore, there exists a convergent subsequence b,,i (0) of g',, (0) converging to some vector v E S X. Then dn(c,,(]K), r
where c,, " R ~
~< rl,
X is the geodesic with b,,(O) -- v.
I-1
As a conclusion one obtains" THEOREM 2.8. l_z,t (X, gl) be a H a d a m a r d manifold and (Y, g2) a H a d a m a r d man~fold with curvature K e2 <~ - k 2 A s s u m e that F" X ~ Y is a quasi-isometric embedding. Then F induces a continuous injective map F ~ 9 X ( o c ) ~ Y(cx~), defined by F..~ ([,']gc,,d) -- Q - ' ([ F o C]qua~i). COROLLARY 2.9. Let (X, gl) and (Y, g2) be two H a d a m a r d man~folds such that K~ i <~ - k 2 < O f o r i E {1,2}. If X and Y are quasi-isometric, X ( o ~ ) and Y(o~:~) are homeomorphic. Note, that the universal coverings of compact homotopy equivalent Riemannian manifolds are quasi-isometric. This even holds in the case of geodesic length spaces, i.e., complete inner metric spaces. More precisely, one has: PROPOSITION 2.10. Let ( M i , d t ) a n d (M2, d2) be two c o m p a c t homotopy equivalent geodesic length spaces and assume that their universal coverings X t a n d X2 exist. Then, with re.wect to the lifted metrics, X t and X2 are quasi-isometric. COROLLARY 2.1 1. Let ( M t , g l ) a n d (M2, g2) be c o m p a c t Riemannian manifolds o f negative curvature with isomorphic f u n d a m e n t a l groups. Then the ideal boundaries X I (r a n d X2(oo) o f their universal coverings are homeomorphic. PROOF. Note that compact K(zr, 1) manifolds Ml, M2 with isomorphic fundamental groups are homotopy equivalent. I-1
G. Knieper
484
A further application, due to Gromov [40], shows that geodesic flows on compact manifolds of negative curvature are orbit equivalent, provided their fundamental groups are isomorphic. One can consider this as a global version of structural stability (see [2]). THEOREM 2.12. Let (Mi, gl) and (M2, g2) be compact Riemannian manifolds o f negative curvature with isomorphic f u n d a m e n t a l groups. Then the geodesic flows dpl,I " S M I --~ SMI and q~tq,2"SM2 --+ SM2 are conjugate (not time preserving). More precisely, there exists a homeomorphism F : SM! ---> SM2 such that
For where T : I~
x
",~2
o F(v),
SMI --> IR is the corresponding time change.
PROOF. L e t F : X! ----> X2 be a lift of a homotopy equivalence and F ~ : Xi (00) ~ X2(oo) be the homeomorphism induced by F. Consider the map G(): S X I ---> SX2 defined by
Go(v) := PF~(,, ).F.,.(,,t )F(rrv), where v + = c,,(+oo) 6 Xi (oe) and for ~, tt 6 X2(oo), we denote by P~.t, the orthogonal projection onto the oriented geodesic determined by ~ and l~. More precisely, P~./~ (x) = b~./~(t) if P , ( x ) = c~.lL(t), where c is a geodesic with c ( - o o ) = ~ and c(+cx~) = tt and P,. is the ordinary orthogonal projection on the geodesic (' introduced above. This map is equivariant and surjective but in general not injective. To obtain a homeomorphism consider the map s : IR x SXI --* I~ defined by
G,,(~'~,,, ~1 -
_d~,<1.,')G()(v) ~
which is an additive cocycle, i.e.,
.,.<~ + t, ~)- .,.(,, ~',,, ~)+ .,.~t, ~). Choose r > 0 such that s(r, v) > 0 for all v 6 SXI and consider the average s(t, v) dt.
r ( v ) -- l"
Define Gr : S X I ---> SX2 by GT(v)-
r
G()(v).
This map is equivariant and, therefore, descends to a map S MI ---> S M2 preserving the geodesic foliations. Furthermore, Gr is injective since
,-/4,/,~~,)c,,(,pl,., v) -,~.~r (4,f~,I v)+.~'(/,v) c,,~,,)
c, (~'I,.,v) -,~.~2 '
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and
m(1)
=
r(dpl~,,v ) + s(l, v) '
- - -1
fo r
s(t
+ l, v)
dt
"t"
is strictly monotone increasing. The monotonicity follows from m , (1) -- -l f t r
s , (t + 1, v l d t = -1( s ( r + I, v) - s(l, v)) -- -1s (r , qS/,o)>O.
l"
1"
Z"
E]
2.3. B u s e m a n n f u n c t i o n s and the horospherical foliations Busemann functions play an important role in the asymptotic geometry and the dynamics of geodesic flows on Hadamard manifolds. They also can be used for an alternative compactification of Hadamard manifolds (see [27] for more details). LEMMA 3.1. Let X be a H a d a m a r d manifold and c : IR --+ X be a geodesic" representing c X (oo) with c(O) = y. Then the limit
b, (x, ~) := lim d ( x , ~'(t)) - t I ---~ %
exists and de[ines a conve~ C2-function. Furthermore, its gntdient has norm I.
PROOF. See I481.
[--]
REMARK.
(a) We remark that the levels of x ~ d(x, c(t)) - t are spheres about c(t). The zero level passes through y. (b) Note, that on any Riemannian manifold ( M , g) the PDE II grad f I1., = I is the Hamilton-Jacobi equation associated to the Hamiltonian G ( v ) ~g(v, v). The integral curves of this gradient field are geodesics parametrized by arc-length. We also remark that CI-functions f with II grad f I1,~,= 1 are of class C I'l , i.e., they have a Lipschitz derivative [31]. DEFINITION 3.2. Let X be a Hadamard manifold. For y ~ X and ~ ~ X ( ~ ) x ~ b,.(x, ~) is called the Busemann function associated to y and ~. The levels of this function are called the horospheres based at ~ 6 X ( ~ ) . We denote by Hor(x.t) the horosphere based at ~ passing through x 6 X. If v ~ S M we denote by Hor,, the horosphere based at c~,(oo) 6 X (c~) passing through c,,(0).
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G. Knieper
REMARK. Since by I (x, ~) - by 2 (x, ~) = by, (Y2, ~) the levels only depend on ~ and not on y. In particular, the gradient grad b v ( x , ~) is independent of y and we denote it by B ( x , ~). For each x, by(x, ~) is the signed distance of x from the horosphere through y. It is decreasing if we move towards ~ and increasing if we move away from ~. From a dynamical point of view the stable and unstable foliations in S X are of particular importance. They are called the horospherical foliations. By W 's and W TM we denote the weak stable and weak unstable foliation, i.e., the foliation of S X whose leaves through v E S X are given by
w ' ' (v) - {w 9 s x I c , , ( + ~ ) = c,,,(+oo) } and w('li ( ~)) -
[ ~l.) E S X l c u ( - (~) ) = c|/,(-(x~))].
Note, if se = c,,(+c~), rl = c , , ( - c ~ ) and y E X, we have W"'(v)-
{ - g r a d b , . ( x , ~ ) I x E X]
and
W'"'(v)- {gradb,,(x, ,1)[x E X} and, therefore, the restrictions 7r:W'"'(v)---> X, zr: W'""(v)---> X of the canonical projection zr : S X -+ X are diffeomorphisms. By W" and W" we denote the strong stable and strong unstable foliation. The leaves through v E SX of those foliations are given by
W"(v)- [-gradb,,(x, ~)[x E Hor,,}, W" (v) -- {gradb,,(x, '1) Ix E Hor_,, },
where ~ = c,,(+oo), r / = c , , ( - ~ )
= c_,,(+o~:~).
The tangent spaces of W " ( v ) and W ' ( v ) can be described as follows. Let S,,(t) and U,,(t) denote the stable and unstable Jacobi tensor along c,, (see Section 1.2, Definition 2.15). Define S ( v ) " - - SI,(O) and U ( v ) " = U,',(0). Note, that U ( v ) and S ( v ) are the second fundamental form for the unstable and stable horosphere. More precisely, U ( v ) : v • ~ v • (v • = {w E T~r~,,)X [ (w, v) = 0 } ) is the symmetric endomorphism with U(v)w--
V,,, gradby (zr(v), q),
where 77= c , , ( - c ~ ) . Similarly, S ( v ) :v • ~ S ( v ) w -- -V~,, gradby (~r (v), ~),
v • is the symmetric endomorphism with
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where r = c v ( ~ ) . Using the decomposition of T SX into horizontal and vertical spaces (see Section 1.1), one has:
and
2.4. Volume growth and entropy formulas Important invariants which reflect the complexity of a dynamical system on an exponential scale are the topological and the measure-theoretic entropies. The topological entropy is an invariant of the topological dynamics and measures in some sense the maximal complexity of the dynamical system. In particular, complexity on small regions can cause positive topological entropy. On the other hand, measure-theoretic entropy is an invariant of the measurable dynamics reflecting the average complexity of the system relative to an invariant measure. If f : X --~ X is a homeomorphism of a compact metric space X, the topological entropy ht,,p(f) is always bigger or equal than the measure-theoretic entropy h1~(]') for any given invariant Borel probability measure 1~. The variational principle (see [2]) implies that for all f-invariant probability measures .h4 ( f ) ht,,p(f) -- sup{h,,(]') ll~ ~ M ( f)}. Measures for which this supremum is attained are called measures of maximal entropy. In Section 5 we will show that certain geodesic flows on manifolds of nonpositive curvature have a unique measure of maximal entropy. Note, that for flows the entropy is defined as the entropy of the time-I map (see [2] or [57,83]) for a comprehensive introduction to entropy theory. In differential geometry there is an asymptotic quantity closely related to entropy. It measures the exponential volume growth on the universal cover of a compact manifold and is defined as follows. Let (M, g) be a compact Riemannian manifold and X be its universal covering. For p 6 X denote by B(p, r) the geodesic ball of radius with respect to the lifted metric. It is a consequence of the compactness of M (observed by Manning [71 ]) that the limit lim r ---* ."y,.)
logvolB(p,r) r
exists and is independent of p ~ X. We denote this limit by h(g) and call it volume entropy. It measures the exponential volume growth of balls in the universal covering. As has been shown by Manning, this number is related to the topological entropy htop(~) of the geodesic flow cpI:SM --~ SM.
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G. Knieper
THEOREM 4.1. Let (M, g) be a compact Riemannian manifold. Then htop(~)/> h(g). If (M, g) has nonpositive curvature then htop (05) -- h (g). REMARK. Freire and Marl6 [36] showed that equality even holds if the metric has no conjugate points. In the case of negative curvature there is an interesting formula expressing the volume entropy as an average of certain curvature quantities. The measure /zH involved in this average is locally of the form d # H = d,k" d/z'", where ,k" is the Lebesgue measure on the strong unstable foliation and/z,.s is the Margulis measure on the weak stable foliation (see Section 3.1). This measure is also obtained as a limit of spherical averages (see [63]). The following entropy formulas are derived in [63]. THEOREM 4.2. Let (M, g) be a compact Riemannian manifold of negative curvature, then h2(g) -- f M Ric(v) - scal o rr + s(v) d/ZH, where Ric and scal are the Ricci and scalar curvature of M, and s(v) is the scalar curvature of the horosphere through v ~ S M. For surfaces the formula specializes to h2(g) - f ' M - K o n d/ztt, where K denotes the Gaussian curvature of M. For 3-dimensional manifolds he(g) - ~'M Ric(v) - scal o Jr d/z//. There are also estimates and formulas for the measure-theoretic entropy on manifolds with no conjugate points. Using Ruelle's inequality and Pesin's formula for the measure entropy of smooth invariant measures [78,79], Freire and Marl6 obtained the following estimate for the measure theoretic entropy of the geodesic flow. THEOREM 4.3. Let (M, g) be a compact manifold without conjugate points and/Z be a ckt-invariant probability measure for the geodesic flow. Let U(v) denote the second fundamental form for the unstable horosphere (see Section 2.3). Then ht~ (~b) <~ f ' M trU d/z. Equality holds if # is a smooth measure.
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2.5. Poincard series and Busemann densities Let (X, d) be a metric space and F C Iso(X) be a discrete infinite subgroup of the isometry group of X. For p, q E X and s E R consider the Poincar6 series e-sd(p,Yq)
P(s, p, q) = Z vEF
For k E I~ consider
a k ( p , q ) = c a r d { y E F Ik ~ d ( p , yq) < k + 1} and define 1
6(p, q) -- k lim (p, q) ---~cy.J -~ a k LEMMA 5.1. 6 is independent of p and q. Furthermore, P(s, p, q) converges for s > 6 and diverges for s < 6. DEFINITION 5.2. The number 6 = 6 ( F ) is called the critical exponent for/-'. The discrete group F is called of divergence type if the Poincar6 series diverges for s = 6 ( F ) . In case that F is a uniform lattice in a Hadamard manifold X, i.e., X / F is compact, we have the following simple criterion for the divergence. LEMMA 5.3. Let M = X / F
be a compact manifold of nonpositive curvature. Then P(s, p, q) diverges for s = 6( F) ~f and only if
L
~ e -'5' vol S(p, r) dr
diverges for one, and, hence, any point p E X. PROOF. Let .T be a fundamental domain for F . Then the divergence of P(s, p, q) is equivalent to the divergence of
f~ ~
e -''l(p'•
dvol(y) =
e -''lIp'.v) dvol(y) --
e - " ' vol S(p, r) dr.
y E l ~"
I-1 REMARK. If M is compact this implies that 6 ( F ) = h(g). In Section 5 we will see that all cocompact actions of F on nonpositively curved spaces are of divergence type.
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490
Let X be a H a d a m a r d space and F C Iso(X) be a discrete subgroup of the isometry group of X. For x e X denote by A ( F ) the accumulation points of the orbit F x in X. This set is given by Fx A X (oo) and is independent of x e X. It is called the limit set of F . If F is cocompact, i.e., X / F compact, then A ( F ) = X(cx~). Next we introduce a certain family of measures on X (cx~), supported on the limit set of the discrete group F which is important from a dynamical and geometrical point of view. This family of measures is called B u s e m a n n densities or conformal densities. DEFINITION 5.4. Let X be a H a d a m a r d manifold and F C Iso(X) be a discrete subgroup. A family of finite Borel measures {/Zp} p~X on X (cx~) is called an c~-dimensional B u s e m a n n density if (a) supp/zp C A ( F ) for all p 6 X, (b) ~dl~q (se) -- e _Otbl , (q,~ ) for almost all ~ e X (oo), (c) {#p}p~X is F-equivariant, i.e., # •
lZp(A) for all Borel sets A Q X ( o o ) .
There is a standard way to construct such a measure. In the case of Fuchsian groups this is due to Patterson [77]. For general hyperbolic spaces it has been intensively studied by Sullivan [82]. We briefly describe this construction. A s s u m e that the group F C Iso(X) is of divergence type. If not, one introduces a positive m o n o t o n e increasing function f :IR + --+ I1~+ such that for each a > 0, we have that f ( r + a ) / f ( r ) ~ 1 as r -+ oo and that the modified Poincar6 series
,~(s, x,
y)- Z f(d(x, yy))e -'''(''• y~l"
diverges for s - 3 ( F ) . Fix x 6 X and for s > 6 and p 6 X consider the measure
e-Sd(p,yx t s p , x , .~' " - -
P(s,x,x)
where 6~. is the Dirac mass associated to y e X. ~tp..,..,. has the following obvious properties. (1) e - ' a / / ' ' ' t <~ p p, ,.,.,.(X) <~e ''lIp''). In particular, the measures are finite. (2) Fx C supp /t p, ,...,. C Fx. Now choose for a fixed p e X a weak limit limk--. ~ kt p,x,.,k =" ktp. The divergence of the series P(s, x, x) for s = 6 and the discreteness o f / - ' yields that the support o f / z p,.,..,k moves toward the limit set if k ~ cx~. More precisely, one obtains" PROPOSITION 5.5. For each q e X, the weak limit l i m , . k ~ lZq,x,.,.k = : #q exists. The family of measure {ktp} p~x is a 8(F)-dimensional Busemann density. REMARK. (a) A B u s e m a n n density constructed in such a way is also called a Patterson-Sullivan measure.
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491
(b) There is up to a constant only one such finite measure if X / F is compact and X has negative curvature, or more generally, is a nonpositively curved rank 1 manifold (see Section 5.1).
3. Horospherical foliations 3.1. Ergodic theory of the foliations In his thesis Margulis [72] constructed for the stable and unstable foliations of an Anosov flow cpt : V ---+ V a measure which is very useful in geometrical and dynamical applications (Margulis measure). Let W'" denote the weak stable foliation of the Anosov flow. Then there exists a family of measures {tt'~"}x~v on the leaves W"*(x) which changes continuously if we slight it along the strong unstable foliations. Furthermore, it contracts exponentially under the geodesic flow on a fixed leave W'' (x). More precisely, cs
/z x o r
t
=e
-hi
cs
#x,
where h = ht,,p is the topological entropy of the flow Ct. Those properties even apply that these measures {~'~:'}.,~v define an invariant transversal measure for the strong unstable foliation. Margulis obtained this measure using a fixed point argument. Bowen and Marcus [15] showed that there exists only one invariant transversal measure for the strong stable foliation. It relies on the Markov property of the foliation. For Anosov diffeomorphisms there is an easy proof due to Fathi, Laudenbach and Po6naru [32]. Since each weak stable leave W'"' (x) is foliated by strong stable leaves, namely W'' (x) = U,~]~ cPt W" (x) one can disintegrate the measure and obtains dtt.'~)"(r y) -- e -h' dt d/~:'i.(y) for all y E W"(x). Similarly one constructs measures {/~i~-"}.,-~v on the weak unstable foliation W'"' with analogous properties, namely lz cu .v
o
~t
--
eht ll, xcu .
Disintegrating [)'.l 9 "" analogous to tti~." yields measures/z.',! on the strong unstable foliation In the case of compact manifolds M with negative curvature one obtains those measures also using the Patterson-Sullivan measure {/zp}p~x constructed in Section 2. Fix a leave W" (v) of the strong stable foliation and consider the bijection P,, : W" (v) --+ X (c~) \ {se }, where ~ = c,,(oo) and P~,(w) = c,,,(-oo). For u, ~ W' (v) define d/]~,(w) "-- dgJrI,,,)(Pv(u')) = e -h/'~''')(rr''''~) dtt~r(,,)(P,,w). This obviously defines a measure as stated above. Using the transformation properties of the Patterson-Sullivan measure one observes that this defines a measure stated above. By uniqueness d~,'i -- d#i,.
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G. Knieper
Using those measures, Margulis constructed a 4~t-invariant measure of maximal entropy (see Section 2.4) which is nowadays called Bowen-Margulis measure and is, therefore, denoted by #BM. Locally #rim is of the form d#xII d ~x' ' 9 Bowen gave another construction of a measure of maximal entropy as a weak limit of measures supported on periodic orbits [13]. Furthermore, he showed [ 14] that the measure of maximal entropy is unique for Anosov flows which, in particular, implies that both constructions lead to the measure. It turns out that the existence and uniqueness of the measure of maximal entropy even holds in the case of geodesic flows on certain compact nonpositively curved manifolds with some amount of hyperbolicity [66]. In Section 5 we study the dynamics of such geodesic flows and give a sketch of the proof of the uniqueness. For a general survey on stable and unstable foliation see [1 ].
3.2. Regularity o f the foliation and rigidity E. Hopf [51] was the first who started to investigate the regularity of the horospherical foliation on a compact surface of variable negative curvature. He obtained that the foliation is of class C I . This was sufficient to apply a "Fubini"-type argument, which is nowadays called the "Hopf argument", to prove ergodicity of the geodesic flow with respect to the Liouville measure. Hurder and Katok [52] proved that for surfaces the foliations are even of class C I'~ll''~-~l which, in particular, implies C I''~ for all c~ < 1. We remark that C I'~ is also deducible from the work of Hirsch, Pugh and Shub [50]. In higher dimensions Green [38] proved C I regularity in the case that the curvature is quarter pinched, i.e., - 4 a ~< K ~< - a for a > 0. This has been also obtained by Hirsch and Pugh [49] using the techniques developed in [50]. In 1966 Anosov showed, for general manifolds of negative curvature (or even general Anosov flows (see [4])) the absolute continuity of the foliation with respect to the Liouville measure. This is still enough to apply Hopf's argument for proving ergodicity. Already Anosov realized [4] that there are obstructions for having high regularity (Anosov cocycle). Those obstructions have been thoroughly studied by Hurder and Katok [521 in the case of geodesic flows on compact surfaces (or more generally for contact Anosov flows on 3-manifolds). They showed that C I'l regularity already implies C ~ . On the other hand, Ghys [37] proved that in dimension 3 contact Anosov flows with C ~ foliations are smoothly conjugate to an algebraic flow, in particular, the Liouville entropy and the topological entropy coincide. For surfaces of negative curvature this implies constant curvature by an entropy rigidity result of Katok (see Section 4, Corollary 1.1). Kanai [55] extended Ghys' result to negatively curved manifolds of higher dimensions. His main tool was a certain connection on the unit tangent bundle called the Kanai connection (for the definition see the discussion below). He showed that the geodesic flow of compact negatively curved spaces (M, g) with C ~ Anosov foliation is time preserving conjugate to a geodesic flow on a compact manifold of constant negative curvature, provided - 9 / 4 < K ~< - 1 . Using deeper dynamical arguments Feres and Katok [34] obtained the same result under the optimal pinching condition - 4 < K ~< - 1. Later on, Feres [33] and Benoist, Foulon and Labourie [9] studied the compact negatively curved manifolds without pinching assumptions. The most general result has been obtained by Benoist, Foulon and Labourie.
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THEOREM 2.1. Let M be a compact manifold of negative curvature and C c~ horospherical foliations. Then the geodesic flow is C ~ - t i m e preserving conjugate to the geodesic flow on a compact locally symmetric space of rank 1. REMARK. (a) Feres obtained the same result under the additional assumption that dim M -4k+2, k ~N. (b) Benoist et al. only need to assume that the foliation is C k for k ~> 2(2(dim M) 2 d i m M + 1). The optimal regularity assumption is probably C 2. This is suggested by the following result of Hamenst~idt [47]. THEOREM 2.2. Let ( M , g ) be a compact manifold of negative curvature with C 2 horospherical foliations. Then the measure theoretic entropy with respect to the Liouville measure coincides with the topological entropy. REMARK. In particular, if A. Katok's entropy rigidity conjecture holds true (see the end of Section 4.1 ) C2-regularity of the horospherical foliation would imply that the mani/bld is locally symmetric. In any case one has" COROLLARY 2.3. Let (M,g()) be a ~'ompact locally symmetric spat'e of negative curvature and g a metric, negatively ~'urved and cot!formally equivalent to g~). Then the Anosov splitting is C 2 if and only if g and g~) ~'oin~'ide up to a ~'onstanr PROOF. The proof follows from an entropy rigidity result of A. Katok (see Corollary 1.8 in Section 4). 1-7 Following Ledrappier [67] we call a negatively curved manilbld asymptotically harmonic if the mean curvature of the horospheres are constant. We recall that harmonic manifolds are those, where the density of geodesic spheres depends only on their radius or equivalently that harmonic functions have the mean value property. Clearly, harmonic manifolds of negative curvature are asymptotically harmonic. Foulon and Labourie [35] used Theorem 2.1 to prove: THEOREM 2.4. Let (M, g) be a compact asymptotically harmonic manifold of negative curvature such that the mean ~'urvature of the horo,wheres are constant. Then the geodesic flow is time preserving conjugate to a geodesic.flow on a compact locally rank 1 symmetric space. Using Theorem 3.1 of Besson, Courtois and Gallot one obtains: THEOREM 2.5. Let (M, g) be a compact manifold of negative curvature such that the horospherical foliation is C ~ or that the manifold is asymptotically harmonic. Then (M, g) is a locally symmetric rank 1 manifi~ld.
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G. Knieper
PROOF. As pointed out above for surfaces one needs only to assume that the foliation is of class C l' l to guarantee constant curvature. Therefore, we assume that the dimension of the manifold is at least 3. According to the results above the geodesic flow is C ~ conjugate to a geodesic flow on a compact locally symmetric space M0. This implies that the fundamental groups of M and M0 are isomorphic and the volume and the topological entropy of the two manifolds coincide. Hence the result follows from the result of Besson, Courtois and Gallot stated above, l--] In all proofs of the theorems stated above a certain connection or a slide variant of it plays a crucial role. It has been introduced by Kanai and is called Kanai connection. We will present a version of this connection introduced by Benoist, Foulon and Labourie. Let (M, g) be a compact n-dimensional manifold of negative curvature, 0 the natural l-form introduced in Section 1.1 and co = dO the associated symplectic structure. Assume that the Anosov splitting (see Section 1.3) T S M = E s G E II 9 E 4~
of the geodesic flow 4)t = S M ---> S M is of class C I. Then there exists a connection uniquely determined by the following three properties. (i) V0 = 0, Vco = 0 and V E" C E", V E" C E u . (ii) Denote by p" : T S M --~ E" and l;i : T S M --~ E 'l the projection onto the stable and unstable bundle. Then V x , X 'l - p " [ X " , X"]
and
V x , , X " - - p " [ X " , X"]
for all sections X 'l ~ F ( E " ) and X" 6 F ( E " ) . dlt-()4)t(v) is the infinitesimal generator of the geodesic flow and (iii) If X ( ; ( v ) X" E F ( E " ) , X " ~ F ( E " ) Vx.X"--[X(;,
X " ] - c~X '
and
Vx~. X" - - [ X ( ; , X" ] + c ~ X ",
where cr -- h t ~ / ( n - l) and ht,~ is the Liouville entropy of the geodesic flow. REMARK. (a) For c~ - - 0 this is the connection originally introduced by Kanai. (b) If I = p" - p ' the bilinear form h ( X , Y) -- w ( X , I Y ) + O ( X ) . O(Y)
defines an indefinite, nondegenerate metric on S M . Since E" and E" are Lagrangian subbundles they are isotropic with respect to h. Property (i) implies that h is parallel, i.e., Vh = 0. However, V is not the Levi-Civita connection of h, since the torsion T is nonvanishing and given by T ( X , Y) -- co(X, Y) . XG + o ( O ( X ) I ( Y )
- O(Y)I(X)).
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(c) If the Anosov splitting is of class C 2 the curvature tensor R of the connection X7 is defined. A computation shows
R ( E " , E " ) - - R ( E S , E'~)--O. In particular, the stable and unstable leaves are flat. The line bundle A t' := A "-I E u of volume forms on E" plays an important role in the rigidity theory of the Anosov foliation. Under the assumption that the Anosov splitting is C 2 the curvature of the induced connection is well defined and given by the flow invariant continuous 2-form F2" (X, Y) = tr R(X, Y)E, for X, Y ~ T S M . Using this fact, Hamenst~idt showed that under the assumption of a C 2splitting the topological and Liouville entropy coincide. For the rigidity Theorem 2.1 of Benoist, Foulon and Labourie an extra ingredient is necessary. In [43] Gromov introduced the notion of rigid geometric structure and proved under certain assumption that the structure is locally homogeneous. As a first step in their proof, the above authors show that such a structure is given by the Kanai connection, provided the Anosov splitting has a certain amount of smoothness.
4. Entropy and rigidity 4.1. Entropy comparison and rigidity Estimates which compare entropy for a pair of metrics on a given compact manifold were first given by A. Katok [64]. He applied those estimates to obtain entropy rigidity results. In particular, for surfaces of higher genus and nonpositive curvature he showed that topological and Liouville entropy never coincide, unless the metric has constant curvature (see Corollary 1.9). In this section we will derive slightly more general versions of his comparison estimates. The proves presented here followed to a large extend Katok's original ideas but have a little more of a geometric flavor. In particular, the notion of geodesic stretch which measures the asymptotic ratio of a pair of Riemannian metrics is very useful. Let M be a compact Riemannian manifold with universal covering X. Fix two Riemannian metrics go and g on M and denote their lifts to X again by go and g. For each vector v in the unit tangent bundle (SM)~ o of go consider the geodesic c,,(s) with respect to the metric go with initial condition b,,(0) = v. If g,, denotes a lift of c~, to the universal covering we define a(v, t) := de (~7,,(0), ~?,,(t)), i.e., we measure the endpoint of the geodesic segment r t]) with respect to the metric g. The triangle inequality implies that a (v, t) is a subadditive cocycle, i.e.,
a v, ,, +
a v, ,,
+ a
v, '2).
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G. Knieper
The following proposition is an easy consequence of the subadditive ergodic theorem [57,
83].
PROPOSITION 1.1. Let/Z be a Cgo-invariantprobability measure
o n ( S M ) g o.
Then
a(v,t) t---, or t
It~ ( g o , g , v) - - lim
exists for ~z-almost all v e (SM)g o. Moreover, we have Iu(go, g, v)d/z = lim t---~ o c t
M )go DEFINITION
M)~()
a(v,t)d/z.
1.2.
I~ (go, g)
"--
fSM)xo
I~ (go, g, v) d/z
is called the geodesic stretch of the metric g relative to g0. The geodesic stretch is bounded from above by the /z-average of the norm II I1,~,= g ( , )l/2 with respect to g (see [64]). More precisely: LEMMA
1.3.
f It, (g(), g) ~< / J( S M
Ilvll,~ d/z. )~(~
REMARK. Using the Cauchy-Schwarz inequality, we obtain
tf
S M ).~'o
)2/ Ilvll,edlz ~
g(v,v)d/z. S M ) x()
The inequality is strict unless g(v, v) -- go(v, v) for/z-almost all v 6 (SM)e(). If/z is a Liouville measure/Z t., we have
S M ),~()
g(v, v) d/z = vole o (M) 9n
tr'e~ g d vole().
The quantity on the right hand side is up to a constant nothing else than the energy of the identity map id" (M, g()) ~ (M, g). The entropy comparison Theorem 1.5 below is based on the following characterization of the measure entropy for an ergodic flow, due to A. Katok [56].
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PROPOSITION 1.4. Let (X, d) be a compact metric space with a continuous flow ~ot : X
X. Suppose lz is a ~ot invariant and ergodic Borel probability measure. Define for T > 0 the metric dT(x. y) -----max{d(qg' (x). qg' (y)) ] 0 ~< t < T}. For c~ ~ (0, 1) and 6 > 0 denote by N(T, 6, or) the minimal number of 6-balls with respect to dT which cover a set of measure >~6. Then h n ,, (~~
log N ( T , 6, c~) ~-,0
T
7"~oo
THEOREM 1.5. Let (M, go) be a compact manifold of nonpositive curvature and lz be a Borel probability measure invariant under the geodesic flow with respect to go. Then for any Riemannian metric g, we have
h(g) >~
I;, (go, g)
h;~ (go)
where h(g) denotes the volume entropy of g (see Section 2.4) REMARK. This theorem is a generalization of a result by A. Katok [56]. He proves under the stronger assumption that go is a metric of negative curvature that
ht,,p(g) >/ f~.SMI~.,, Ilpll.~ d u " hj,(g~)) holds for all metrics g and all probability measures p invariant under the geodesic flow with respect to go. It also generalizes a corresponding result in [64], where it is only proved in the case w h e n / z is the Liouville measure. However, most of the ideas in the proof are already present in Katok's original paper. PROOF. Since every invariant measure has an ergodic decomposition one can assume that p is ergodic. Let f C X be a fundamental domain in the universal covering X and let ~ be a positive number. For T > 0 consider the set
a(v,t)
Ii~ (g0, g) < e , t ~ T } .
The ergodicity of/z implies that/z(Sf).~ 0 \ A T,~:) ~ 0 as T ~ o~. Hence, we can choose T > 0 such that p(AT'~:) >~ 1/2. For t > T and 6 > 0 let S(t, 6) be a maximal subset of A T,~: which is 6-separated with respect to
d,(v. w ) = maxId~,,(c,,(s), c,,(s)) l O ~ s <~t I.
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499
Then the geodesic flows with respect to go a n d g are, up to a scaling, time preserving conjugate. More precisely, there exists a h o m e o m o r p h i s m F : ( S M ) g o --+ ( S M ) g such that F o dpgl,- ~; o F, where c -- h ( g ) / h ( g o ) . REMARK.
(a) This theorem is essentially due to Burger [20]. He proves that the marked length spectra of g() and g are proportional. This implies the existence of a conjugating homeomorphism. See [64] for a different proof. (b) The existence of such a conjugacy implies in dimension two that g() and g are isometric up to a constant (see Theorem 5.1 of Section 4.5). The next result follows easily from Theorem 1.5 and Lemma 1.3. COROLLARY 1.7. Let (M, g()) be a compact manifold o f nonpositive curvature a n d lZ be a dp.e() invariant probability measure. Then, f o r any other metric g the estimate
h t~ (go)
h ( g ) >~
(1.1)
f~ S'M t~,, II v I1,,~dl~ holds, l f ~ = lz 1. is the Liouville measure,
h(g)2 >~ I
vole()(M) h t, i. (g()) 2 JM tr~(~g d vol~(~
(1.2)
If additionally g is confi~rmally equivalent to g(), we have
h(g) )
( vol~() M ) I/,, vol~ M
ht'c (g())"
(1.3)
Itg() is not a flat metric inequalities (1.2) and (1.3) are strict, unless g and g() are identical up to constant. REMARK.
(a) If (M, g()) is a compact manifold of nonpositive curvature, the metric g() is nonflat iff the Liouville entropy hni (g()) > 0 is positive. This is a consequence of Pesin's entropy formula given in Section 2, Theorem 4.3. (b) It turns out that the last inequality is true for metrics not necessarily conformally equivalent to g(), provided g() is a locally symmetric metric of negative curvature (see Section 4.3). As a consequence, we obtain the following entropy rigidity result due to A. Katok [56] for a conformal class of a locally symmetric space of negative curvature.
G. Knieper
500
COROLLARY 1.8 (Katok). Let (M, go) be a locally symmetric compact manifold of negative curvature. If g is a metric of nonpositive curvature, conformally equivalent to go with volg M = volg 0 M, then
h(g) >~ht~,. (g). Furthermore, equality holds if and only if g = go. PROOF. Since go is a locally symmetric metric of negative curvature p,/~ is the measure of maximal entropy, i.e., the Bowen-Margulis measure. Using that g is conformally equivalent to go with the same total volume, we obtain from (1.3) in Corollary 1.7, h(g) ~ h(go), where equality holds if and only if g = go. Applying Corollary 1.7 again, we conclude that h(go) >~hl~ (g). [-q This yields for surfaces: COROLLARY 1.9 (Katok). Let M be a surface of a genus >~2. Then for all Riemannian metrics g ~?fnonpositive curvature
h(g) > h!~1 (g), unless g is a metric ~?['constant negative curvature. PROOF. The proof follows from the uniformization theorem which states that each Riemannian metric on M is conformally equivalent to a metric of constant negative curvature. [--I It is a well-known conjecture due to A. Katok [ 18] that Corollary 1.9 should generalize to higher dimensions in the following sense. CONJECTURE (Entropy rigidity). Let (M, g) be a compact manifold ~?[negative curvature.
Then h(g) = h / , / ( g ) , if and only if(M, g) is locally symmetric.
4.2. Regularity of topological entropy As we have seen in the last section the entropies provide interesting functionals on the space of Riemannian metrics. Those functionals are never determined by the local geometry of the manifolds unless we are in special symmetric situations. The next theorem shows that the topological entropy is smooth on a certain class of nonpositively curved metrics. Originally this has been proved in [59] for negatively curved metrics. However, the
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501
theorem remains true for nonpositively curved metrics provided the measure of maximal entropy is unique. As we will see in Section 5, the uniqueness holds for rank 1 manifolds, i.e., manifolds of nonpositive curvature with some amount of hyperbolicity. THEOREM 2.1. Let (M, g0) be a compact rank 1 manifold and g z , - e <~ )~ <~ e, a perturbation within the space of metrics of nonpositive curvature. Then )~ e--> h(gz) is a C I -function and its derivative is given by d d)~
h(g())
h (g~) --
2
f,s'
M)~,)
d
dJk
gz (v, v) d/~g,,,
where/~() is the measure of maximal entropy f o r the geodesic flow 4)~(). PROOF. Using Corollary 1.7, we obtain
1 h(g())
./ISM)x( ' Ilvll~ dlz~,,
- 1) ~< h ( g z ) -
h(g())
IIv I1~,, d/z e>. - 1) .
Linearizing )~ ---> IIv ll,~,>, at ) ~ - 0 implies
II v I1,> - II v I1,,, + 9~
d
Ilvll.~, + o(~, v). s =()
Substituting this in the above inequality, we obtain for suitable o t(~), o2(~.)"
( f
h (g()) -;k
d
SM)~,,, ds
Ilvll,~, dlzo + ol (~.)) s =()
<~ h(gx) - h(g()) <~ h(g()) -)~
,S'M)~;,
ds
II vtl,~, dlz,~; + o2 ( k ) ) . s =()
Each weak limit of (lZq,~) for X -+ 0 will determine a measure of maximal entropy for g(). Since this measure is unique in the rank 1 situation (see Section 5.5) X --~ #,~>, is continuous in the weak topology. Consequently, we obtain the desired formula for the derivative. F1 Using symbolic dynamics the authors prove in [58]" THEOREM 2.2. Let (M, g()) be a compact manifold with a C~-metric of negative curvature. Let X ---> g ~ , - ~ <~ )~ <<.e be a C~-perturbation then the function )~ e--> h(g~) is a C ~-function as well.
G. Knieper
502
The result has been extended to the Liouville entropy and even more generally to pressure by Contreras [22].
4.3. Minimal entropy A celebrated theorem by Besson, Courtois, and Gallot [10] states that inequality (1.3) in Corollary 1.7 remains true for all metrics g provided go is a locally symmetric metric of negative curvature on a compact manifold. In fact, it even holds if g is a metric on a manifold with fundamental group isomorphic to the fundamental group of the locally symmetric space. We call this theorem, which was also conjectured by Gromov [41], the minimal entropy theorem. THEOREM 3.1. Let (M0, go) be a compact locally symmetric space of negative curvature. Assume that (M, g) is another compact Riemannian manifold such that rcl (M) ~- rcl (Mo). Then (vol~,, M) !/'' h(g) >~ (vole M)l/'' h(go). /f dim M~)= dim M ~> 3 equality holds if and only if g and g~) are homothetic, i.e., are isometric up to a constant. The rigidity part of this theorem immediately implies the Mostow rigidity theorem for compact locally symmetric spaces of negative curvature. COROLLARY 3.2. Let (Mo, gl)) and (Mi, gl ) be two compact locally symmetric spaces of negative curvature and of dimension strictly bigger than 2. Then gl and g() are homothetic provided the fundamental groups of Mo and MI are isomorphic. Now we will give a proof of Theorem 3. l under the assumption that (M, g) is a compact manifold of negative curvature. In this case, the use of the Patterson-Sullivan measure constructed in Section 2.5 simplifies the proof considerably (see [ 1 1]).
Step 1 (Construction of the homeomorphism between the ideal boundaries). Let M0 = (X/Fo, go) and M = (Y/F, g) be compact homotopy equivalent manifolds of negative curvature. Then the homotopy equivalence q9 : M --+ M0 lifts to a quasi-isometry ~ : Y --~ X which is equivariant with respect to the F-action on Y, i.e., ~ov
= A(V) oq3
for all V 6/-', where A : F -+ /-]) is an isomorphism between the fundamental groups F of M and/]) of M0. Furthermore, ~ induces a homeomorphism F : Y ( ~ ) --~ X ( ~ ) between
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503
the ideal boundaries of Y and X which is equivariant with respect to the F-action on Y, i.e., Foy=A(y)
oF.
PROOF. It is a general fact that a homotopy equivalence between compact manifolds lifts to a quasi-isometry between the corresponding universal coverings (see 2.10). Furthermore, Corollary 2.9 of Section 2 implies that the quasi-isometry 99 : Y --~ X induces a homeomorphism F : Y (cx~) ~ X (cxz) between the ideal boundary. The /-'-equivariance of the quasi-isometry implies the/-'-equivariance of F. IN Step 2 (Construction o f the natural map). In Step 2 we construct, using the barycenter of the Patterson-Sullivan measure, a differentiable/-'-equivariant extension f : Y -+ X of the homeomorphism F : Y ( ~ ) ~ X ( ~ ) . The /-'-equivariance of f yields that f can be viewed as a map from M to M0.
LEMMA 3.3 (Barycenter of a measure). Let X be a Hadamard manifold o f negative curvature and let xo E X be a fixed reference point. Then, f o r any finite measure )~ on X ( ~ ) , the function q : X ~ IK given by
q(x)-
fx
(,,x:)
bx,,(x, ~) dX(~)
is smooth and depends only up to an additive constant on xl~. Moreover, tfsupp ,k = X (cx:~) then q is strictly conv~:~ and there e~ists a uniquely determined absolute minimum S = S()~) E X which is called the bao'center o f'/k.
PROOF. The smoothness of q follows from the smoothness of the Busemann function x --~ bx~,(x, ~). Furthermore, the equation b~,(x, ~) - b,~ (x, ~) = b~,,(xi, ~) implies that q only changes by an additive constant if we choose another reference point X l. It is a consequence of the convexity of the Busemann function that q is strictly convex if supp)~ = X(cx~). Using again that ,k has full support, we obtain that q ( x ) --~ cx~ as x --+ ~1 E X(~:~). Hence, q has a unique minimum. [-1 LEMMA 3.4. Let M = ( Y / f ' , g ) and M~ = ( X / ~ ) , g o ) be two compact homotopy equivalent manifolds o f negative curvature and F : Y ( c x ~ ) ~ X ( ~ ) be the induced homeomorphisms between the ideal boundaries. Let {lZ v } vE Y be the Patterson-Sullivan measure o f Y and S be the barycenter-map introduced in Lemma 3.3. Then the map f : Y ~ X given by .
.
f (y) = S ( F . l z v ) is a smooth F-equivariantfunction, i.e., f o y ( y ) = A ( y ) o f (y) f o r a l l y E Y and y E F.
REMARK. Following Besson, Courtois and Gallot [ 11] we call f the natural map.
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504
PROOF. Since for each y e Y f (y) e X is the unique critical point of the map
g"(x ' F(~)) d/~y (~) Xo 0 , ~')d(F,#y)(~') -- fy q(x)= fx(oo) X~:~ (oo) it is for each y e Y the unique solution of the equation O = G(x , y)"= fx
gradt,t"X() g"(x , F ( $ ) ) d # ~' (~) 9 (oo)
Furthermore, the map G ' X x Y --+ T X is differentiable and for each fixed y e Y the covariant derivative of the vector field x ~ G(x, y) on X induces a nondegenerate linear map Vg~ y)" TxX ~ TxX given by t,,g0(x , F ($)) d# v($) V,~'G(x, y)= fx(~) V~'}' grad u.r() Hence, the implicit function theorem is applicable and implies the smoothness of f . The F-equivariance follows from the F'-equivariance of the Patterson-Sullivan measure and the fact that the barycenter-map commutes with isometries. More precisely, for all y e F' and y ~ Y we have:
f (y y) -- S(F,/t•
= S( F, y, pt ,.) = S( F o y),/t,.
= S(A(y),F,#>.)- A(y)S(F, lt,.)- A(y)f(y).
i-1
Step 3 (The derivative of the natural map).
As usual, the implicit function theorem also provides a formula for the derivative of the implicitly defined map. To simplify the notation in our computation, we denote for any Hadamard manifold (Z, h) the gradient grad b~l,,(z, r/) of the Busemann function b h:0(z, r/) by Bh(z, 77) (see the remark in Section 2.3). Now, fix a point y0 e Y. Using the transformation rule of the Patterson-Sullivan measure, it follows that the natural map f : Y --+ X fulfills the equation
O-- fx
( ,"~ )
B'~"(f (y), F($))e -hl'~l'.'''l'''~ du,.,,($).
In order to compute the derivative of f , we choose a differentiable curve t ~ y(t) e Y such that y(0) -- y and #(0) -- v. Then, differentiation of the equation above yields" D dt
B.~"(f(y(t)), F($))e -h/~)h:'~',,(-''(')'s) d/t,.,, ($) t --() i X (~'x~)
B.~"(f(y), DI'( v)v -- fx ( ~ ) V'U~ 9
F($)) dkt,.($)
.
- h(g) f x ( ~ )
B.~~
F(~))(B,~(y,$), v)d#,,(~).
(3.1)
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505
i be the normalised Patterson-Sullivan measure. Consider for Let /.iv = ~,,,(x(oc))/Zy x 9 X and y 9 Y the endomorphism Qx,y: Tv X --+ Tv X given by
Qx,y(W) - f x
V~'l'Bg('(x, (ec)
F(~)) d/2,,(~)
(3.2)
and the linear map
Lx,v : T r Y ---->T r X given by
Lr.y(V)
-
fx(o~)Bg~
-
F(~))(B'~(y,~),
v)d/2r(~).
Then Equation (3. l) can be written in the following compact way: (3.3)
Q.f(y),,, o D f ( y ) = h(g)Lf(.,,),y.
Denote by U(,.,t) the second fundamental form of the horosphere corre ~ponding to x E X and 71 E X ( ~ ) . Then for all u~ E T~ X, we have
V',~?B'~""(x, F(~))= U(,.t.(~,,(u,-
(u,, B'V"(x, F(~')))B'V"(x, F(~))).
Since U(.~-.F(,~)) is symmetric and positive definite, the map Q.,-,~.: T~.X ~ symmetric and positive definite as well and, therefore,
D r ( y ) -- h(g)QTl,.)..,, o
L i(.,.i.,..
T~-X is
(3.4)
Step 4 (Estimate o f the Jacobian of the natural map). Let w,vo and to v be the volume forms associated to the Riemannian metrics g(I and g. Define the Jacobian of .f to be Jac f ( y ) = det D f ( y ) , where the determinant of a linear map is defined via the volume forms. Therefore, Equation (3.4) implies Jac f ( y ) = h" (g) LEMMA 3.5. For all x
det L.I. ( v ). v det Q f(y i. v
9X and y EY, we have
I det L.,.,.,. I ~< ~
1
(det H,.. ,.) I
/2
where Hv, r : Tv X --+ Tv X is the symmetric endomorphism given by
Hv.y(W)
"
-
-
fx(~)(,,,, 8.,,, (x,
8.",, (x,
506
G. Knieper
REMARK. It would be of considerable interest to know under which assumptions I det Lx,yl is bounded from below by a positive constant. In general, due to the existence of exotic structure, we cannot expect the existence of such a constant. PROOF.
The Cauchy-Schwarz inequality implies for all w 6 Tr X and v 6 Tv Y
I(Lx,xV,
w),e,, I -
fx(~) (Bg"(x, F(~)), w)
<~(Hr,y(W), W)'/2 fx
(ec)
v)d/iv(~)
(B'e(y,~), v)2d[zv(~).
(3.5)
Now choose an orthonormal basis el . . . . . e,, which diagonalizes the symmetric endomorphism Hr,v :TrX --+ TrX. Then H x , y ( e i ) - - l z i e i , and since IlB,e~ F(~))llgo = 1 and/2v is a probability measure, the definition of H~-,v implies ,tZ i > 0 and Y'/i~l/zi = 1. Since the lemma is trivially true if det L r, r = 0, we assume det Lx, v ~ 0. In this case, Lx,y is invertible, and we can apply the G r a m - S c h m i d t orthonormalization process to the basis L -.~,y I (e j). If bl . . . . . b, is the resulting orthonormal basis in Tv Y, we have .J
L.v,v(bj) -- Z cijej, i=1
i.e., the associated matrix is upper triangular. Using (3.5),
ej)l
IdetL.,.,.I.j= I
~<
lZl (H,.v(ej) , e.i) I/2
.j = I
(B'e(y,~),bj .j = I
d/~v(~)
(~)
1
Since the geometric mean is bounded from above by the arithmetic mean, we obtain
]det Lx' xv l <~ (det" H~""v ) l /2 ( -nl f i f j =I = (detH~. v)l/2
(B g (y, ~), bj) 2 d/~ v(~-)) ''/2
1
nn/2 9
I-1
This implies the following estimate for the Jacobian of f :
[Jac f(Y)l ~< h"(g)
(det Hf (y), y) I/2 1 det Q.f(y),v " n"/2"
(3.6)
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Step 5 (Proof o f the minimal entropy theorem). Only in the last step we will assume that the target manifold (M0, g0) is a locally symmetric space of negative curvature. To simplify the calculation we will assume that (M0, go) is a manifold of constant curvature - 1. Then the second fundamental form of the horosphere is the identity and the endomorphism Qx.y :Tx X ~ Tx X defined in (3.2) is given by Qx. v(w) = w - H~. r(w). I f / z l . . . . . /z,, are the eigenvalues of Hf~y).,., we obtain, using (3.6), the following estimate for the Jacobian of f :
IJacf(Y)l ~
1-I.';= I /zl/2 ./ h"(g) l-I.'~=l (1 - / z j )
9
1
(3.7)
nn/2 9
So far, we did not make use of the assumption n ~> 3. Since for surfaces of higher genus the Teichmtiller space of nonisometric metrics of constant negative curvature is nontrivial, the rigidity part of the minimal entropy theorem cannot be true for n -- 2. It is an amusing fact that the assumption n ~> 3 will be only used in the following innocent looking lemma (see [10]). LEMMA 3.6. Let n be an integer strictly bigger than 2. Let lzi . . . . . lz, be real numbers such that 0 < lz i < 1 and }--~.'i=_I/zj - 1. Then i-iit.i=i/zj!/2
( l / n ),/.,-
<~
H
l-I i = l ( 1 - / t j
)
( 1 - 1/n)"
where the inequality is stri~'t unless I~l -- 1~2 . . . . .
I~,, -- l / n .
The minimal entropy theorem will be a consequence of the following proposition. PROPOSITION 3.7. Let (MI), gr be a compact n ~ 3-dimensional symmetric space o f negative curvature and ( M , g) be any compact manifold o f negative curvature homotopy equivalent to Mo. Then, f o r all y ~ M, the following estimate f o r the Jacobian o f the natural map f : M --+ M~) holds: [Jac f ( y ) [ ~<
h"(g) h" (gt))
Furthermore, this estimate is strict unless I,~go) D f (y) is an isometry.
PROOF. We will prove this proposition under the assumption that (Mo, go) has constant curvature - 1. The inequality is a consequence of Lemma 3.6 since
IJacf(y)l ~< and h (g0) -- n - 1.
l--I';= I I~.jt /2
1
h" (g) i--l,}=, ( 1 _ lt.i ) n"/2
<~
h"(g)
h"(g)
(n - 1)"
h" (go)
0,.,~
~ ~ ~.,-i
.wl
;;~
Z ~
ov,-i
9
i
~,.,."
I
~
II
II
I II
=:
9
......,
II
E
m
;
m
.'-~
t-.
~
~
~
r162
%//
~
~..,
t ~176
......
II
~
~ ,.,,q
o
~
"~
,..,
,d
.~
~
=
Z
,...,
~-
,,I I
Hyperbolic" dynamics and Riemannian geometry
509
where L := Lf(y),y, we obtain from (3.8) by choosing an orthonormal basis bl . . . . . b,, Tv Y I1
t r A - - n 2 h2(g) t r L t L - - n 2 h2(g) E ( L b j , h 2(go) h 2(go) j= I ~< n2 h2(g) 1 ~ h2(go) n
j= I
fr"
Lbj)
h2(g) (BZ (y' ~)' bj)2 d/2v(~)= n ~ h2(go) (~)
Hence, (tr--~A)"~<(h2(g)) ' ' - h2(g(,) n
h2"(g) h 2, (go)
= det A
and, therefore, h2 (g) id.
A - D f (y)' . D f ( y) = h2(g() ) This implies h2(g) --(v.
h2(g~)
v).~- - ( D r ( y ) ' D f ( y)v, v ) - (Df ( y)v, D.f ( y)v).~,, ,
i.e., ~hI'e~D.f(.v) is an isometry.
I-q
Now, the minimal entropy theorem follows quite easily from Proposition 3.7. Namely, since f has degree 1, we obtain
vol(M(),go)--fM u'.e,,--fM f* wg. -- fM Jac.f (y) u,z 1)
h" (g) fM h" (g) vol(M g) <~ h" (gl~------~ u,e -- h" (gr ' " The inequality is strict, unless Jacf(y) -- h" (g)/h" (g(~) for all y E M. In this case f is an homothety. QUESTION. Does the above estimate remain true if (M, go) is a compact locally symmetric space of nonpositive curvature and higher rank? 4.4. Spectral rigidity Let (M, g) be a compact Riemannian manifold and denote by Ag "C~(M) --->C ~ ( M ) the Laplace-Beltrami operator with respect to the metric g. A smooth perturbation gx, - s ~<
510
G. Knieper
~< e of the metric g = go is called isospectral if the spectra of A gx and A g o coincide for all ~ 9 l - e , +e]. A Riemannian manifold (M, g) is called spectrally rigid if all isospectral perturbations gz of g are trivial, i.e., gz = qg~g, where ~0z'M ~ M is a one parameter family of diffeomorphisms. The following theorem, due to Croke and Sharafutdinov [25], generalizes previous work by Guillemin and Kazhdan [44,45], and Min-Oo [73]. THEOREM 4.1. A compact Riemannian manifold o f negative curvature is spectrally rigid. The first step of the proof is that the Ag-spectrum of a compact negatively curved manifold ( M , g ) determines the length spectrum A ( g ) C Ii~+ of periodic geodesics (without multiplicity) on M. This is a simple consequence of the trace formulas due to Duistermaat and Guillemin. Let s be the set of free homotopy classes of M. If g is a metric of negative curvature we call the map Lg :S-2 ~ R + which associates to each free homotopy class the length of the unique closed geodesic representing this class, the marked length spectrum. COROLLARY 4.2. Let g z , - e <~ X ~ e, be an isospectral perturbation o f a negatively curved metric g = go on a compact manifold. Then Lg~ = Lg o fi~r all )~ 9 l - e , +e]. PROOF. This follows from the continuity of)~ ~ L ez (c~) and since L ez (a) is contained in the countable set A(go) C IR+. Hence, L ~ (c~) is constant for all c~ 9 s [-] REMARK. In particular, for surfaces the theorem follows from the work of Croke [23] and Otal [751 (see Section 4.5). Theorem 4.1 is now a consequence of the rigidity of the length spectrum. THEOREM 4.3. Let (M, g) be a compact manifold o f negative curvature and g x , - e ,k <~ ~, be a perturbation such that A ( g z ) = A(go). Then the perturbation is trivial.
<~
The first step in the proof of this theorem consists of the following simple observation: LEMMA 4.4. Let gz be a perturbation as above and fl -- if2 [z=z~gz. Then fi~r each closed geodesic c:[0, ~1 ~ M in (M, gz~), we have
Using this simple lemma, there are two main ingredients for the proof of Theorem 4.3. The first one is due to Liv~ic [68]. The theorem asserts that to each function f 9 C ~ ( S M ) which integrates to zero along each periodic trajectory there exists a function F : SM ~ II~ such that X~; F = f , where X c ( v ) = (v, 0) is the infinitesimal generator of the geodesic flow (see Section 1.1) (the smooth version of this theorem is due to de la Llave [69] et al.). The second main ingredient which distinguishes the work of Croke and Sharuftdinov from previous approaches is a certain formula of Weitzenb6ck type on the tangent bundle
Hyperbolic" dynamics and Riemannian geometry
511
of a manifold. This formula has been discovered by Pestov and Sharafutdinov [8 1] and is called Pestov's identity. We will give an coordinate free presentation of this formula (see the Appendix for a proof). To state this formula, we introduce the following notations. If q9 9 C e~ ( T M ) , we denote by gradqg(v) -- (grad h qg(v), grad v qg(v)) 9 T v T M the gradient of 99 with respect to the Sasaki metric. The vector fields grad h 99(v), grad v qg(v) 9 TjrI,,)M are the associated horizontal and vertical components. A smooth map X : T M ~ T M is called a semibasic vector field if it preserves the fibers of T M, i.e., n" o X = n'. The horizontal and vertical gradient are examples of such maps. For a semibasic vector field there is a natural notion of the horizontal and vertical divergence, div h and div v (see Appendix). THEOREM 4.5 (Pestov's identity). For all ~o e C ~ ( T M ) , 2(grad h 99(v), g r a d V ( X ~ ; q ) ( v ) ) ) -
[Igrad"
we have
II2 + d i v h y ( v ) + divVZ(v)
- ( R (grad v qg(v), v) v, grad" q9( v )), where Y(v)-
(grad h 99(v), gradVqg)v - ( v , grad h qg(v)) grad v g)(v)
and Z ( v ) -- X~;qg(v) 9grad h 99(v) --(v, grad h 99(v))grad h qg(v).
Let C , , ~ ( T M ) -- {q9 E C ~ ( T M ) I qg(~.v) -- )~'"qg(v), for all ~. > 0} the homogeneous functions of degree m. Then we obtain the following integrated version of Pestov's identity. THEOREM 4.6. Let M be a compact n-dimensional Riemannian man~['old and q9 E C,,~ ( T M ) . Then
- 2 L M ~Pdivh gradV (X~;qg) dlz t.
-- LM ]]gradh ~p(v)II2 d/tt.
+ (n + 2m) f (XG~p(v)) 2 d/,L
-- L'M (R (gradV qg(v), v)v,
grad v ~o(v))d#t.
G. Knieper
512
REMARK. This formula, together with Liv~ic's theorem, immediately shows: If M is a compact manifold of negative curvature and f 6 C ~ ( M , R) satisfies fc f o zr dt -- 0 for all closed geodesics c, then f = 0. Namely, if f -- XG~0 for a function ~o 6 C ~ ( S M , Ii~), the left hand side of the identity above is zero since grad v f -- 0. But then f -- Xcq9 = 0. Now we want to generalize this to symmetric m-tensor fields. Denote by F m ( M ) :-- F ( Q m T * M ) the C ~ - s e c t i o n s in the m times covariant tensor bundle (m-tensor fields) and S m ( M ) = F ( @ ~ y m T * M ) be the subbundle of symmetric tensor fields. Denote by V : F m (M) --+/-,,,,+ l (M) the covariant derivative, induced by the Levi-Civita connection V and given by (Vc~)(X, Xl . . . . . Xm) = (Vxc~)(Xl . . . . . Xm) I11
= X ~ ( X i . . . . . Xm) - ~
~ ( X l . . . . . X i - I , V x Xi, Xi+l . . . . . X,n)
i=1
for vector fields X, Xi
7
9 S'"
M and c~ E F'" (M). If we restrict V to S'" (M), we obtain
on
m * M) . ( M ) ---> F ( T * M | 6~ ._.~ymT
Now we compose V with the symmetrization cr 9 F ( T * M @ .-.sym ~'" T ' M ) --+ S '''+l ( M ) and obtain an operator
* : S'" ( M ) ~
S'" + I (M).
The Riemannian metric g induces an inner product ( , ) integration induces an inner product ( , ) on the sections
on the fibers of |
and
(c~, fl) -- f M (C~, ~) d vol. The adjoint of g* with respect to ( , ) given by got = -tr1,2
is the negative divergence g : S '''+l (M) --+ S ''z (M)
Vc~.
In particular, if Ei . . . . . E,, are orthonormal vector fields we have for all vector fields X I . . . . . Xm that I1
Sc~(XI .....
X,,,) =
-
~-'~(VEiot)(Ei, Xl . . . . . X,,,). i=1
One obtains the following decomposition of symmetric tensor fields.
Hyperbolic dynamics and Riemannian geometry
513
PROPOSITION 4.7. Let ( M , g) be a c o m p a c t R i e m a n n i a n manifold a n d ot e S m ( M ) be a C~ o f the symmetric tensor bundle. Then there exist sections fl ~ S m - ! ( M ) , g ~ S m ( M ) such that 8y = 0 a n d c~ = 6 * f l + y.
This decomposition is unique if m is even. I f m is o d d it is unique provided (fl ' | e where g is the R i e m a n n i a n metric and ~ = (m - 1)/2.
= 0,
REMARK. The restriction for odd m is necessary since V ( | eg) -- 0 by the parallelity of g. To apply Pestov's identity in the case of symmetric m-tensor field one needs the following observation. LEMMA 4.8. Let o t e S'" ( M ) be a symmetric tensor field a n d F ( v ) = a ( v . . . . . v). Then, ]'or all v e T M,
m6ot(v . . . . . v) -- - d i v h grad v F ( v ) and X(; F ( v ) -- 5 * a ( v . . . . . v). PROOF.
d
(grad v F (v), 77) --
ot(v + tO . . . . .
v + tO)
-
-
mot(o, v .....
v).
1=(}
Note, that for a semibasic vector field X the horizontal divergence is given by
div h X ( v ) --
(X (V,,i (t)),
7
e.j ),
t--l)
j= I
where ei . . . . . e,, is an orthonormal basis of T~rl,,)M and v,,j (t) is the parallel translation of v along the geodesic c,.i. Hence,
- d i v h grad v F ( v ) -- m
kd
-~
j = I
ot((ei (t), v,,i (t) . . . . . v,,i (t)) l
I!
=
m Z ( V e i c ~ ) ( e . / , v . . . . . v) ./= I
=
m 9~u.
The second equality follows since d
X~; F ( v ) -- -~
u(r t--O
.....
r
v) -
(v~,u)(v .....
v) - a * u ( v . . . . .
v).
I-1
G. Knieper
514
Now we can prove the following theorem due to Croke and Sharafutdinov [25]. THEOREM 4.9. Let (M, g) be a c o m p a c t negatively c u r v e d R i e m a n n i a n m a n i f o l d a n d ot ~ S m M a s y m m e t r i c tensor field. I f the integral
fo
e a ( b ( t ) . . . . . b(t)) dt
is equal to zero f o r all closed geodesics c" [0, g.] --+ M, there exists a s y m m e t r i c tensor f i e l d fl ~ S m - I ( M ) such that (~ * ~ ~
Ol.
I f m is even, fl is uniquely determined.
PROOF. The smooth version of Liv~ic's theorem implies the existence of a smooth function qg : S M ---> R such that XGcp(v) = a ( v . . . . . v). Extending r to a function on T M such that r 6 C ~ ( T M ) is homogeneous of degree m - 1, we have XGcp(v) =c~(v . . . . . v) for all v ~ T M . Decomposing c~ according to Proposition 4.7, we obtain: X(;q9 = ~ = •*/3 + y. By L e m m a 4.8 6*fl(v . . . . . v) = X ( , f l ( v . . . . . v) and, therefore, X(, (r - / ~ ) = y. By the same lemma div h grad v X(;(q9 - [4) - div h grad v y -- - m S y -- 0. Using the integrated version of Pestov's identity we deduce X(; (99 - / 4 ) - - 0 which yields the claim. F1 To prove the spectral rigidity we need to consider the special case of symmetric two forms.
COROLLARY 4.1 0. Let M be a c o m p a c t m a n i f o l d a n d )7 ~ S 2 ( M ) be such that )1 integrates to zero along all closed geodesics. Then there exists a smooth o n e - p a r a m e t e r f a m i l y o f d i f f e o m o r p h i s m s r " M ---> M such that d )1-- -~
~0;g. l--()
PROOF. By the above theorem there exists a 1-form/4 such that r/-- 6*[3. Let Z be the vector field dual to 2/4. Then one obtains for all vector fields X, Y on M d
r
~ * ~ ( X , Y) -- g ( V x Z, Y) -~- g ( V y X, Z) -- -~
Y),
t =0
where qgt is the one parameter subgroup of the diffeomorphism group associated to Z.
VI
Hyperbolic" dynamics and Riemannian geometry
515
4.5. Rigidity o f the marked length spectrum In [18] it is conjectured that the marked length spectrum is rigid within the class of compact negatively curved manifolds, i.e., two metrics of negative curvature on a given compact manifold are isometric if their marked length spectra coincide. According to the last section, the infinitesimal version of this conjecture is true. For surfaces the conjecture has been proved independently by Croke [23] and Otal [75] using different methods. THEOREM 5.1. Let M be a compact surface o f genus ~ 2. If g l and g2 are metrics o f negative curvature with the same marked length spectrum, then g l and g2 are isometric. REMARK. This result has been extended by Croke, Fathi and Feldman [24] to the case of compact surfaces of nonpositive curvature. The above conjecture can be reduced to the question whether a C~ preserving conjugacy between geodesic flows is induced by an isometry [46] (see also [64]). More precisely" PROPOSITION 5.2. Let gl and g2 be metrics o f negative curvature on a compact manifold. Then the marked length spectra o f g i and g2 coincide if and only ~f the geodesic" flow o f the metrics g l and g2 are C~)-time preserving c'onjugate. REMARK. Note, that a time preserving C~
(SM)~ and ( S M ) 2 is a homeomorphism F ' ( S M ) I - - >
of the geodesic flows 4~tR,4~ on ( S M ) 2 such that 4)~ o F - - F o 4)I~.
5. Ergodic properties of weakly hyperbolic spaces In this chapter we summarize some of the recent results on the dynamics and asymptotic geometry of nonpositively curved spaces (see [65] and [66]). We are in particular interested in spaces with a certain amount of hyperbolicity which we will call weakly hyperbolic or rank 1 spaces. The rank of a nonpositively curved manifold has been introduced by Ballmann, Brin and Eberlein [7]. It measures the amount of flatness of a manifold M of nonpositive curvature.
5.1. Definition o f rank and rank rigidity DEFINITION l . l . For a unit tangent vector v ~ S M , rank(v) is the dimension parallel Jacobi fields along the geodesic c,,. The minimum of rank(v) over all v ~ called the rank of M. If rank(v) = rank(M) the geodesic c,, and the corresponding v = b,,(0) are called regular. The set of regular vectors v ~ S M is called the regular
of the S M is vector set.
REMARK. (a) Since s is always a parallel Jacobi field and since the dimension of parallel fields is not bigger than dim M we have 1 <~ rank(M) ~< dim M.
G. Knieper
516
(b) A parallel perpendicular field along a geodesic c is a Jacobi field if and only if the sectional curvature of the planes spanned by E ( t ) and b(t) is zero for all t 6 I~. (c) It is easy to see that rank (Mi x M2) -- rank (Mi) + rank (M2). (d) For symmetric spaces X of nonpositive curvature the rank is defined as the maximal dimension of a totally geodesic flat subspace. In the case of symmetric spaces both notions coincide since each geodesic is contained in a maximal flat. If the geodesic cv is contained in precisely one maximal flat F it follows that rank(v) - - d i m F = rank(M). In particular, c,, is regular. A fundamental result in the theory of manifolds of nonpositive curvature is the rank rigidity result of Ballmann [5] and Burns and Spatzier [1 9]. THEOREM 1.2. Let M be a compact manifold o f nonpositive curvature with irreducible universal covering X. Then the manifold either has rank 1 or X is a symmetric space o f higher rank. Theorem 1.2 implies that most compact manifolds of nonpositive curvature are rank l manifolds. We note that they do in general not admit a second metric of negative curvature. There are compact rank 1 manifolds which have Z k, k ~> 2, as a subgroup of their fundamental group. By Preissmann's theorem this excludes the existence of a metric of negative curvature. Nevertheless, we will see that the geodesic flow of such manifolds still exhibits some of the dynamical properties similar to those which are well-known in negative curvature, or more generally for Anosov flows. Therefore, we call rank 1 manifolds also weakly hyperbolic spaces. Unlike for Hadamard manifolds of negative curvature, two points at infinity cannot always be connected by a geodesic. However, in the neighborhood of rank 1 geodesics the geometry behaves very much like in negative curvature. For n 6 1~1we denote by d,, the metric on the unit tangent bundle SX defined by d,,(v, w ) = max{d(c,,(t), c,,,(t)) I t c [0, n]}. Then we have: LEMMA 1.3. Let X be a Hadamard manifold and c be a rank 1 geodesic on X. For each e > 0 there are neighborhoods U o f c ( - o o ) and V o f c ( + o o ) such that f o r all ~ ~ U and ~ V there exists a rank 1 geodesic h connecting ~ and 71 such that dl (i'(0),/~(0)) ~< e. In particular, according to the flat strip theorem, h is uniquely determined up to parametrization. Furthermore, axial isometries fixing a rank 1 geodesic behave like in negative curvature. By definition, an isometry y 6 Iso(X) is called axial if there exist a geodesic (axis) c and to > 0 such that y ( c ( t ) ) = c(t + to) for all t 6 It~. If X / F is compact all deck transformations y 6 F are axial.
Hyperbolic dynamics and Riemannian geometry
517
LEMMA 1.4. Let X be a Hadamard manifold and c" IR ~ X be a rank 1 axis o f y 9 Iso(X). Then f o r all neighborhoods U o f c ( - e c ) and V o f c ( + e ~ ) there exists no 9 N such that f o r all n >~ no m
V" ( X \ U) C V
D
and
y - " ( X \ V) C U
f o r all n ~ no. REMARK. Both lemmata imply that the endpoint of a rank 1 axis can be connected to any other point in X (oo) by a rank 1 geodesic. There are many rank 1 axes if X / F is a rank 1 manifold of finite volume. LEMMA 1.5. Let X be a rank 1 Hadamard manifold such that X/I-" has finite volume. If c" N ~ X is a rank 1 geodesic, then there exists a sequence o f rank 1 axes c, such that Cn ~
C.
PROOF. First one observes that in the case of finite volume the geodesic flow is nonwandering (Poincar6 recurrence). This implies that the endpoints of a geodesic c" R X are F-dual, i.e., there exists a sequence y,, 9 F such that for one p 9 X (hence, for all p 9 X) y,, p ~ c(+
F~ - x (oo). PROOF. If 71 9 X(oo), 71 # ~, and V C X(oo) is a neighborhood of 71, we can choose a rank 1 axis c such that c ( + o o ) 9 V and c ( - o o ) r {71,~}. Choose a neighborhood U C X (o~) of c ( - o o ) which does not contain ~. If y 9 F is the axial isometry associated to c, Lemma 1.4 implies the existence o f n 9 N such that y" (X (cx~) \ U) C V. In particular, ?,"~ 9 V. V1 In Definition 5.4 of Section 2.5 we introduced certain families of finite Borel measures {/z/,}/,cx on X (oo) called Busemann densities. They play an important role to understand asymptotic properties of the geometry and dynamical properties of the geodesic flow. In the next section we will study those measures for rank 1 manifolds. It turns out that they are uniquely determined [65].
518
G. Knieper
5.2. Uniqueness o f the Busemann density First we will show that on rank 1 manifolds X with compact quotient M -- X / F a family {#p}pEX of Busemann densities has certain semi-local properties. For ~ 6 X(c~) and p 6 X consider the projections prs " X ~ X (oo)
and
prp" X \ {p} ~ X (oo)
along geodesics emanating from se and p, respectively. Hence, pr~(x) -- C~,x (cx~), where c~,c is the geodesic with c ~ , x ( - c ~ ) = ~, c~,x(O) -- x and p r p ( x ) = cp,x(cx~), where cp,x is the geodesic with cp,x(O) = p, Cp,x(d(p, x)) = x. PROPOSITION 2.1. Let {lZp }pEX be an or-dimensional Busemann density. (a) Then there are constants R > O, g. > 0 such that lzp(prx B(p, p)) ~ g f o r all x ~ X (oQ) U X, where B ( p , p) is the open geodesic' ball o f radius p >~ R about p. (b) Furthermore, there is a constant b - b(p) such that f o r all x E X and ~ =
,:'/,..,-( - o e ) . -]e -ud(p,.v) <~ IJp(pr~(B(x, p ) ) ) <~h e -"'i(t''') b
(c) A similar estimate holds 0" we project from p E X, namely there is a constant a - a (p) > 0 such that 1 -e a
-~"'P'-') <
PROOF. The estimates in (b) and (c) follow from (a) and the defining properties o f / z p . Namely, if A C X (cx~)
~ p ( A ) = fA e-~'t"(P"l) d/zx(r/).
If A = pr~ B(x, p), or p r p B ( x , p) then I(bx (p, r/) - d ( p , x))l is bounded by a constant for all 77 E A. This yields (b) and (c). The first estimate is a consequence of the following steps. STEP 1. supp/zp -- X (oo) f o r one and, hence, f o r all p E X since I" acts minimally on x (oo). STEP 2. For each ~ E X ( o o ) and x E X there exists r > 0 such t h a t p r ~ ( B ( x , r)) contains some open sets.
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To see that, choose a rank 1 axis c on X. By the remark following L e m m a 1.4 the endpoints of such a geodesic can be connected to all other points of X(cx~). For 6 X ( o c ) let he., 7 be a connecting geodesic with h~.,l(+oo) = r / = c ( + o o ) . Since he., 1 is a rank 1 geodesic, L e m m a 1.4 implies that for E > 0 and x = x0 -- h~.,l(O) there exists an open neighborhood U of r / s u c h that U C p r ~ ( B ( x , e)). If x is arbitrary one chooses r = d(xo, x) + e. For v ~ S~ X = 7r-I (x) and e > 0 consider the open neighborhood
C~;(v) = {c,,,(~) I w 6 S ~ X a n d / ( v , w) < e}. Fix a compact set K such that U • v ( K ) -- X and a reference point x() E K. Then the following uniform version of Step 2 holds. STEP 3. There exist R > 0 a n d ~ > 0 such that f o r all p ~ K and x E X
C~:(v) C p r x ( B ( p , R ) ) f o r some v ~ S~-()X. Suppose Step 3 is false. Then there are sequences p , 6 K, x, ~ X such that
Ci/,,(v) Cpr,,,(B(p,,,n)) for all v ~ S~)X. Since x, c Int B(p,,. n) implies pr~,, ( B ( p , , n)) = X (cx~), we can assume after choosing a subsequence that x,, ~ ~ and p,, ~ p ~ K. Then Step 2 implies the existence of r > 0, e > 0 and vr ~ S,~)X such that C~:(v()) C p r E ( B ( p , r ) ) . The continuity of the projection implies the existence of nr such that for all n ~> nr we have: C~./2(v()) C prx,, ( B ( p , , r)). But this contradicts the choice of the sequence if l / n < e / 2 and n > r. The next step follows from the positivity of #/, on open sets.
STEP 4. For all ~ > 0 there, exists a constant g. -- f ( e ) > 0 such that
u,(c
(v)) > e
f o r all v E S,.oX and p ~ K. m
Now consider x 6 X and p E X. Choose y 6 F such that y p ~ K. Since
# p ( P r x ( B ( P , P))) -- # • 2 1 5
P))
the estimate ( l ) follows from Steps 3 and 4. Choose p ~> R and a reference point x() 6 X. Then we call
8,,.'
- prx,,
p))
71
G. Knieper
520
a ball of radius r = e -d(x~ about se = prxo(X) at infinity. For A C X(cxz) and m ~> 0 consider the m-dimensional Hausdorff measure
H m ( A ) - - ,-+o lim inf
rjm rj <~e, A c_ j=!
B~ (~j ) j=l
associated to the "ball"-structure introduced above. Then part (b) of Proposition 2.1 can be used to prove that an c~-dimensional B u s e m a n n density is equivalent to the c~-dimensional Hausdorff measure. More precisely, there exists a constant c > 1 such that 1 - lzxo (A) <~ H ~ ( A ) <~ c lzx,, ( A ) . c
Since there exists an 6 ( F ) - d i m e n s i o n a l Busemann density by Patterson's construction,
ot - - 6 ( F ) = h --dimHausd. X(oo). Furthermore, the estimate implies that all B u s e m a n n densities are equivalent. One even can show that the F - a c t i o n is ergodic. This implies that the density is uniquely determined (see [65] for more details).
5.3. Volume growth and growth rate of regular closed geodesics The local estimate of the Busemann density in Proposition 2.1(c) can be used to obtain some asymptotic properties of the geometry (see [65]). THEOREM 3.1. Let (M = X / F , g) be a compact rank 1 manifold and x ~ X. Then there exists a constant a > 1 such that 1 vol S,.(x) -~< <~a a
e hr
for all r > O. PROOF. Choose p ~> 2R, where R is as in Proposition 2.1 and let x I . . . . . x,,, be a maximal p-separating set in S,.(x), i.e., a set of maximal cardinality such that d ( x i , x j ) >/p for i g: j . In particular, [,_J B(xi, p) D_ Sr(x) and B(xi, p/2) 71 B(xj, p/2) = 0 for i g: j . The local property of the B u s e m a n n density implies 1 -hr
-e b
~ #x(Prx(B(xi, fi) 71Sr(x))) ~ b e - / "
for all t3 with 2R ~> ,6 I> R and a constant b > 0. In particular, !11
,.,.
-''r i=1
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and
-1m e -hr ~ #x b
I UPr.,. !11 t
"tBrxi, p / 2 ) n Sr(x))
t
~/s
i=!
and, hence,
1 eh r <~ m <~ fle hr
/3
for a constant/4 > 1. From the uniformity of the geometry follows 1
for r I> r0, R <~/5 ~< 2R and, therefore,
Ill vol S,.(x) <. Z
vol(B(xi, p) N S,.(x)) <<.fig e h'
i--I
and 1 eh ,. vol S,.(x)>~ ~'" vol(B(xi, p ) N S,.(x))>~ [4---[ i=!
which yields the claim.
I-1
REMARK. We remark that in the case of negative curvature the method of the proof can be used to derive Margulis' result, namely the existence of the limit
lim r-~ ~
w)l S,. (x) = a (x). e h,
The reason is that in this case we can work with balls B(xi, p) of arbitrarily small size. Furthermore, the function a ( x ) can be interpreted as the total mass of lzx. COROLLARY 3.2. If M = X / F is a compact rank 1 manifold, the Poincard series
Z e-s,l(l~,• diverges for s - h f o r all p, q ~ X, i.e., F is of divergence type.
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522
PROOF. Let .T" C X be a fundamental domain
f~. z e-sd(p'Yq) dv~ ycF
-- y ~ fy --
Since the right hand side diverges for hence, for all p, q ~ X.
fo ~ s =
e -sd(p'q) dvol(q) (.9:)
e -sr vol Sr (p) dr. h,
the Poincar6 series diverges for one and, IN
REMARK. In [65] it is shown that all discrete cocompact lattices of Hadamard spaces are of divergence type. In the following we study the growth rate of regular closed geodesics on rank 1 manifolds. In the case of negative curvature all geodesics are regular and one can use symbolic dynamics and the analytic theory of zeta-functions to obtain much better estimates (see [3]). Now let M = X/F be a compact rank 1 manifold. We are interested in counting primitive (prime) non-oriented closed geodesics, i.e., closed geodesics which are not an iterate of another one. The flat strip Theorem 1.5 in Section 2 implies that the regular closed geodesics are uniquely determined in their free homotopy class. In general this is not true for the singular closed geodesics, since typically, as the example of a flat torus shows, they come in an uncountable free homotopic family. Therefore, we select a maximal set 79(M) of primitive and geometrically distinct closed geodesics representing different free homotopy classes. We denote by 79reg(M), respectively 79sing(M) :-- 79(M) \ 79tog(M) the subset of all regular, respectively singular closed geodesics. Furthermore, 79(t), 79,.cg(t), respectively 79~ing(t) is the subset of geodesics in 79(M), 79,-eg(M), respectively 79.~i,lg(M) of length ~< t. Finally, P(t) = card 79(t), P,-eg(t) = card 79,-cg(t) and &ing(t) = card 79.~ing(t) denote their corresponding cardinality. It is easy to see that under the assumption of nonpositive curvature free homotopic closed geodesics have the same length. Therefore, the counting functions above are independent of the selection 79(M). The free homotopy classes are in one-to-one correspondence with the conjugacy classes of F. Therefore, the set 7a(M) is in one-to-one correspondence with the following equivalence classes in F \ {id}. Two elements YI, Y2 ~ F \ {id} are called equivalent if and only if there exists n, in E Z and r/E F such that •
=
Denote by [F] the set of equivalence classes. We call Y ~ F \ {id} primitive if it cannot be written in the form y = ~'", m t> 2 for some 13 ~ F . Each [y] ~ [F] can be represented as [y] = {r/yr rl -! I Y0 ~ F, Y0 primitive, r/~ F, n ~ Z}. For y ~ F' denote by g(y) - - i n f { d ( p ,
YP) I P E X}
Hyperbolic dynamics and Riemannian geometry.
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the length of y. Since our manifold is compact, there exists p0 9 X such that d(p0, yp0) = g(y) and the geodesic through po, Ypo is an axis of y. The projection onto M is a closed geodesic of length (prime period) g([y]) -- min{g(r/) I r / 9 [y]} = g(Yo), where Y0 is a primitive element representing [y]. Therefore,
P(t) -- card{[y]
9
IF] I e([•
~< t}.
LEMMA 3.3. There is a constant al > 1 such that P(t) <<.al e hr. PROOF. Fix a fundamental domain .T. Then each [y] 9 [F] can be represented by a primitive element Yo with axis passing through 3m. If g(Yo) ~< t then yo(.T') _ Bt+,.(xo) for some constant c > 0. Since Theorem 3.1 implies vol(Bt+,.(xo)) <~eht/~, for/~ > 0, we obtain P(t) <~/7)eht/vol(f') = a e ht . I-7 In Section 5.6 we will improve this trivial upper bound. For that we will use the uniqueness of the measure of maximal entropy derived in Section 5.5. In this section we study the regular closed geodesics. First we will obtain a better upper bound for P,.c~(t). To do this, we are following an approach which is often used in trace formulas. Define for p, q 9 X
a(l,,q,t) = card{y
9
F ld(p, y q ) <~ t}.
If Xl~.tl: IR ~ I~ is the characteristic function of the interval [0, t ], we can write
a(p, q, t) = Z XI'"tI(d(P' Yq))" yel"
LEMMA 3.4. Let .T be a fundamental domain of'X/N then
f
a(q, q, t)dvol(q) ~< vol B(p, t + D),
where D = diam f and p 9 f . PROOF. Choose p e .T. Since d(p, vq) <~d(p, q) + d(q, yq), we have that a(q, q, t) <~ a (p, q, t + D). Hence, /,
f F a ( q , q, t) dvol(q) <~]~ a(p, q, t + D) dvoi(q)
=
E XI~ ycl"
Yq))dvol(q)
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524
-- • -
LEMMA 3.5.
-
Xl~
q))dv~
fx Xl~
q ) ) d v o l ( q ) = vol
Br+D(p).
D
There is a constant b > 0 such that
O(t)= Z
g.(c) <~be h'.
cc P,.~g(t) PROOF. The lemma above, together with Theorem 3.1 implies the existence of a constant b l > 0 such that bl 9ehr ~> •
= where Conj(M) Conj(y) be the F/Zy, the map representing the
f.,. XlO,~l (d(x, yx)) dvol(x)
E
E
orEConj(M) rE~,
denotes the set conjugacy classes of F. Let Z~, be the centralizer and conjugacy class of y E /-'. If R• C /-' is a set representing the cosets R• -+ Conj(y) with ~1 ~ 71-1Yr/is a bijection. Choose a subset H C / - ' conjugacy classes Conj(M). Then we obtain
h,. eht E E fT x.,,.,,(,(x.,,-I y 71x)) dvol(x) y E I7 tIER ),
=zz/,, y E H tIER),
Xl~).rl(d(x, yx))dvol(x) ([f')
XlO.rl(d(x, y x ) ) yEFI
where f ( Z • is a Z• -- {yl')' In 6 Z} follows since each would bound a flat
dvol(x),
(Zy )
fundamental domain of the centralizer Z• of Y. If Y is regular, then is the infinite cyclic group generated by a primitive element Y0. This element/4 of Z• fixes the regular axis a• of V, since otherwise a• strip. Furthermore, a fundamental domain of Z• is of the form
exp,,yo(,) N~yo(,)--. f • O~
U
where
N,,n~(.,) is the
normal space to the axis a•
U• (5)- Ix E .Y• I d(x, a• <~~}
in a•
Let
Hyperbolic dynamics and Riemannian geometry be the 8-tubular neighborhood of a•
g(Yo)] in f'•
525
For each x E U•
d(x, yx) <~d(x,xo) + d(xo, yxo) + d(yxo, yx) <~2~ + g.(y), where we choose x0 6 a• f(Y0)] such that d(xo, x) <<.8. Denote b y / 7 o the subset of primitive elements i n / 7 and Ho(t) the elements in H() of length smaller than t. Then
bl
>~
XiO,tl(d(x, yxl)dvol(x)
>~
E
~
y E F / o ( T - I)
=
Xlo,tl(d(x, yx))dvol(x) y(l/2)
E
vol U•
yEH()(t- I ) By comparison with Euclidean geometry, we obtain a constant b2 > 0 such that vol U •
~> f ( y ) . b2.
Note, that the set of primitive elements corresponds to the regular closed geodesics 7~,.eg( M ). Hence,
E
b3eht>~
f(Y)--
yEH~(t-
I)
E
f(~')
~'E'P,.c,., (t - I)
DI
for a new constant b3 > 0 which yields the lemma. PROPOSITION
3.6.
There exists a constant a, > 0 such that a)
P,-c,g(t) <~ - e hI. t
PROOF. Define O(t) -- ~,.E.p,~Vtt ) f(C). Let li > 0 be smaller than the length of the shortest closed geodesic. Then
Prc,,(t) .
' d0(s)
ft~ .
.
.
S
O(t) + f' O(s) e hI ~ ds <~ b - - + h t
S-
fl~ t eh'
t
( e ht ) ds=O -7- " K]
Now we are going to obtain a lower bound for the number of regular closed geodesics. Define Frog = {y 6 F I the axis corresponding to y is regular}.
526
G. Knieper
If y 6/"reg then obviously [y ] C Freg. Therefore, if [ Freg] C [/"] are the equivalence classes consisting of regular elements, then Preg(t) -- card{[y] 6 [Freg] I s
~< t}.
LEMMA 3.7. Let s be the length o f the shortest closed geodesic in M, B r ( p ) a ball o f radius r -- s / 4 about p ~ X and c'II~ --+ X be an axis o f the primitive element Yo ~ F. Then card{y IV =/3Y~/3-1 , e(y) ~< t, /3(c) fq B r ( p ) # O] ~< t/r. PROOF. If g(Y0) -- to and y -- r -I with e(y) ~< t then Ikl ~< t/to. If k is fixed, k ~ 0,/~ ?'~/3-I # / ~ y ~ / ~ - I implies /~-I/3 r Z• where Z• is the centralizer of yo. In particular, /4(c) r fi(c) since otherwise /~-~/~ -- y~ for g 6 Z. If/4(c) and /~(c) have nontrivial intersection with B r ( p ) we will find q, ~ 6 c([0, to]) and n , m E Z such that /3y0" (q), r, ~ ,,, implies d ( q , ~) >~.3/2 since #Yo,,, (q) 6 B,.(p). Then g --/3y 0,, -~ ~ --/~V0 .3/2 ~> d ( g ( q ) , ~, (~)) >1 d(~, -I gq, q) - d(~, q) >~s - d(~, q). Hence, for fixed k there are not more than 2to/s different elements of the form y --/-J yCf/~-I such that/4(c) fq B,.(p) ~ 0 and the lemma follows. [--1 Essentially, the same estimate holds if we replace B,.(p) by any compact domain.
COROLLARY 3.8. If K C X is a compact set and c'II~ ~ element )/~ ~ F. Then there exists a constant e such that card{y I v --/JY(~/4-~, g(Y) <<.t, [4(r
X an axis o f the primitive
B,.(p) ~ O} <~ et.
In order to obtain a lower bound for P,-eg(t), we will construct many regular elements in F such that the corresponding axis are passing through a compact domain. For that we need some preparations. For p E X and 0 ~< r < t define ~ " ( p ) = {y ~ F I r < d ( p , Y ( P ) ) <~ t} and ~ ( p ) = ~()(p). Then, for fixed r/> 0, Theorem 3.1 implies card C"(P) ~> ~eh' for a constant b > 0. Moreover, we need the following two lemmata. Consider the metric dl on S M defined in Section 5.1 and denote by B I (v, r) the ball of radius r about v E S M with respect to di. LEMMA 3.9. Let c be a hyperbolic: axis q f rl ~ neighborhoods U o f c ( - o o ) , V o f c ( + o o ) and all Y ~ I-" with d(xo, yxo) <<.t and Y U f-) V -- 0 endpoints in U and V. Moreover, h(O) ~ B I (~'(0),
I-" and x o - c(O). For e > 0 there are constants p > O, n ~ 1~ such that f o r the axis a o f ~"Y~" is hyperbolic with e) and f(O"yrl") <~ t + p.
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527
LEMMA 3.10. Let c be a regular axis o f rl 9 F. I f xo = c(O), there are constants r, k > 0 such that card{y 9 l-'t+r(XO) l y U n V--91} ~ l/4cardFtk(xo). The proofs of Lemmata 3.9 and 3. l 0 can be found in [62]. THEOREM 3.1 I. There are constants a3 > 0 a n d to > 0 such that Prcg(t) ~> a3. e h t / t
forallt
> to.
PROOF. If M is a compact rank 1 manifold the regular closed geodesics are dense. Choose a regular axis c : R ---> X and define for e > 0: A~:(t) "-- {V 9 F I f ( y ) ~< t the axis a of ), is regular, h(0) 9 B I (~(0), e)}.
It follows from Lemmata 3.9 and 3. l0 that card A~.(t + q) >~ 1/4card Ft k (x0), where q = p + r. Corollary 3.8 yields card{[ylOA~(t)J~
Preg(t) >/
card A ~ ( t ) eh I >/a3 /t et
for all t ~> to.
[-I
REMARK. Since the regular closed geodesics are dense, the above argument shows that the number of regular closed geodesics passing through any open set grows exponentially. More precisely, if O C S M is any open subset in S M and p..eg(t )o _ card{c Ic closed, primitive, and regular, g(c) ~< t, ~' n O -r ~4} then Pr~~ ( t ) >1 b e t ' ' / t
for a constant D > 0 depending on O.
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528
5.4. Construction of an ergodic measure of maximal entropy Now we construct, using the Patterson-Sullivan measure # p , an invariant measure for the geodesic flow (see [66]). Let G be the space of nonparametrized geodesics in X and
be the set of points in X (cxz) • X (e~) which can be connected by at least one geodesic. According to its definition, # p is F-quasi-invariant with R a d o n - N i k o d y m cocycle
f ( Y , ~) --e-hi'P(•
p'~). For (se, r/) ~ g7E consider
flp(~, rl) = -(bp(q, ~) + bp(q, rl)), where q is a point on a geodesic c connecting ~ and r/. In geometrical terms fl/,(~, 17) is the length of the segment c which is cut out by the horoballs through (p, ~) and (p, 7/). Since gradq bp(q, ~) = - g r a d q bp(q, rl) for all points on geodesics connecting ~ and rl, this number is independent of the choice of q. An easy computation shows: LEMMA 4.1. For p ~ X d/~(~, r/) = e/'~'1~''1) d#/,(~) d#/,(r/)
defines a F-invariant measure on G/" Let P : S X --+ ~/': be the projection given by P(v) = ( c , , ( - ~ ) , c,,(cx~)), where c,, is the geodesic with ~',,(0) = v. Note that the projection 7 r : S X ~ X maps the fibers of P onto geodesics or flat totally geodesic submanifolds of X. Therefore, the flow acts isometrically on the fibers of P. Then ~ induces a 4~t-invariant measure # on S X by setting for each Borel subset A C S X #(A)-
f~7, vol(Tr (P - t (8, '1)A A))d/](~, 0),
where vol is the induced volume element on the submanifolds. By the F-invariance of this measure descends to a flow-invariant measure on S M. We normalize it so that it becomes a probability measure, i.e., # (SM) = I. In order to compute the entropy of the measure # we consider a measurable partition .,4 = {A I . . . . . Am} of S M such that the diameters of all elements in .,4 are less than e with a / ' ) , then, by the definition respect to the metric dl introduced in Section 5.1 9 If v 6 a 6 ~,,/, of .,4, it follows that It--I = c
A k=()
8,,, (e
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529
LEMMA 4.2. Let 0 < e < min{R, inj(M)}, where inj(M) is the injectivity radius o f M.
Then there is a constant a > 0 such that #(or) <~ e - h " a f o r all ot E A(n) ~4~ 9 PROOF. Since cr C j('-3nu,=0I ~-k Bj~ (q~kv, e) if v E cr we have for all w E ot that d ( c v ( t ) , c,,,(t)) ~< e for all t E [0, n]. Let p 6 X be the reference point used in the definition of the measure/_z and fi E S X be a lift of v such that d(rrfi, p) ~< diam M = r0. Since e < i n j ( M ) we can lift the set c~ to a set 6~ e S X such that for all tb e (~ we have d(c,~,(t), c~,(t)) <<.e for all t E [0, n]. Let c~(n) = x and ~ = c,~,(-o<~). Let C~.x be the geodesic connecting and x such that c~,,. (n) = x. Then,
d(c~,x(O), p) <~ d(c~,x(O), 7r(ffo)) + d(yr(ffv), p) <~ 2e + r(,, i.e., ~ E p r x ( B ( p , r() + 2e)). Therefore, if P : S X ~ we have
P(a) C
U
{'l}
•
~!~: denotes the endpoint projection,
pr,,(B(x,e,)).
qCpt'~ ( B(p.r()+2~'))
For each 71E p r x ( B ( p , r() + 2e)) choose a point q E B ( p , r() + 2~) that lies on the geodesic c,~.,. Then, using the transformation rule for the Busemann density, Proposition 2.1 and the estimate d ( q , x) ~> d ( x , 7r~) - d(zr ~, p) - d ( p , q) >>.n - r(i - (2e + r0), we obtain
l~t,(pr,,(B(x, e))) <~ eh'l(t"q)lzq(pr,,(B(x, e))) ~ eh('"'+e~;)be-h"(q") <~ [~e -h'' for a constant/~ > 0. Since l z v ( p r x ( B ( p , ro + 2e))) ~< ~ l , ( X ( o o ) ) < cxz the proof follows from the definition of it. Fl THEOREM 4.3. The measure lz is a measure r h~ ( ~ ) = h ( ~ )
entropy f o r the geodesic flow, i.e.,
=h.
PROOF. Choose a partition A as above. Then by L e m m a 4.2
H(A~'))= ~ /z(c~)(-log/z(c~))>/(hn-loga)Z /~(ot)--hn-loga. EA(") c r . , "4,
~
EA(") " "4,
Hence, h ( ~ , A) ~> h. The theorem follows since h >~ h1,(dp) >~ h ( ~ , A) >~ h.
F-l
530
G. Knieper
Now we show the ergodicity of/z. For v ~ S X and ~ = cv(oo) let W" ( v ) -
{w 6 S X I w -- - g r a d b ~ r v ( q , ~ ) , b ~ r v ( q , ~ ) - - 0 }
denote the strong stable manifold associated to v. PROPOSITION 4.4. Let v E S X be a recurrent regular vector. Then f o r all w E W s (v)
a~ (0' (~), , ' (~)) --, o as t --+ cx~.
PROOF. Since v is recurrent there exist sequences y,, E F and t,, --~ ~ with Dv,,(ckt"(v)) ~
v
as n ~ ~ .
If w is asymptotic to v the function t ~ dl (4~~(v), 4~~(w)) is m o n o t o n e decreasing. If it does not converge to 0, there exists a constant bl such that dl (4) t (v), 4~t (w)) ~> bl for all t ~> 0, and we conclude b, ~< d,
(~,,,+, (v), 4,,,, +, (u,)) <~ d l ( v , u ,)
for s E l - t , , , cx~). Therefore, b, <~ d l ( 4 9 ' D y , 4 f l " ( v ) , 49' D y , r
<~ d , ( v , u,).
Passing to a subsequence if necessary we can assume that Dy,, (ckt"(w)) ~ then we obtain
w' E S X . But
h, <~ d~ (r (v), r (to')) <~ d, (v, w) for all s 6 I1~. Hence, 4r' (v) bounds a flat strip by the flat strip theorem contradicting the fact that v is regular. [-1 Let a be a rank l axis. By L e m m a 1.4 there exist open neighborhoods U, V C X (cxz) of a ( - o o ) , a ( + ~ ) such that each ~ E U and r/E V can be connected by a rank 1 geodesic c which, therefore, is unique up to parametrization (see [6] for details). Define G,-,~c(U, V) - {c geodesic I i" recurrent, c(-cx~) E U, c(+cx~) E V }. This set has full measure in the set of all geodesics with endpoints in U and V with respect to /z. Let f : S X ~ R be a continuous F-equivariant function. Then, by the Birkhoff ergodic theorem,
f + (c) "-- "r~o~limTI ~ 7" f ( ~ ( t ) ) d t
Hyperbolic dynamics and Riemannian geometry
531
exists and is equal to
l f, 7. f ( 6 ( - t ) ) d t
f-(c)'=
lim -T~ T
for/x-almost all geodesics c. Note that f + and f of c. Consider the set ~rcc(U, V ) = [c 6 ~rcc(U, V) l f + ( c ) -
are independent of the parametrization
f-(c)}
which has full measure in {7,-c,:(U, V). By the structure of the measure # there exists a geodesic cl 6 {Tree(U, V) with cl ( - c o ) -- ~ such that G~ -- {r/6 V
I c ( + ~ ) = o, c(-oo) - ~; and f + (c) - f - ( c ) } c V
has full measure with respect to/zp. LEMMA 4.5. f + is constant a.e. on G,.cc(U, V). PROOF. For almost every c ~ O,-c~.(U, V) we have c ( + ~ ) 6 G~. Let c2 be the geodesic such that c 2 ( + ~ ) -- c(+cx~) and c2(-cxz) -- ~ -- t'l ( - ~ ) . Then Proposition 4.4 implies: lim d,(~'2(t),~'(t))--
lim
d,(~'2(t).~',(t))--O.
The continuity of f implies that f+(c)
-
THEOREM 4.6.
f+(c2)
-
f-
(c2) -
f-
(el) -
D
f+(cl).
The measure lz is an ergodic measure with re,wect to the geodesic flow
on SM. ,-,.,
PROOF. Let f ' S X ~ IR be a continuous F-invariant function and V -- {71 6 V 171 = c.(+eo), c 6 {7,.co(u, v)}. It follows from L e m m a 4.5 and Proposition 4.4 that f + is constant a.e. on the set of geodesics c with c ( + o o ) E V. Consider the set
yEP
We know that Y A V has full measure in V with respect to ~;, and, hence, with respect to #q for any q. Every ~ E X(c~) has an open neighborhood V(~) that can be moved into V by an element of F . Therefore, Y n V (~) has full measure in V (~) for each ~ 6 X (co) and, hence, Y has full measure in X ( ~ ) . Consequently, the set Z = {c I c ( + ~ ) 6 Y} has full measure with respect to #. Then, by the F-invariance of f + , f + is constant a.e. on Z. Hence, the measure/z is ergodic with respect to the geodesic flow. I-1
532
G. Knieper
COROLLARY 4.7. The regular set in S M has full measure with respect to #.
PROOF. The discussion above shows that the regular set has positive measure. Since the regular set is flow-invariant, the corollary is a consequence of the ergodicity of/z. [--1 In the next section we sketch a proof of the uniqueness of the measure of maximal entropy, obtained in [66].
5.5. The uniqueness o f the measure o f maximal entropy Following Bowen [12] we call a h o m e o m o r p h i s m T: V ~ V on a compact metric space (V, d) entropy expansive if its local topological entropy vanishes. More precisely, for x E V and e > 0 let Z~:(x) = {y E V I d ( T " x , T " y ) <~ e for all n E 1~1}is the e-neighborhood of the orbit {T"X},E's If h ( T , Z~:(x)) = 0 for all x E V we call T entropy expansive with expansivity constant e. Under this assumption, for any T-invariant probability measure v, h,,(T) = h,,(T, A ) provided A = {Ai . . . . . Am } is a Borel partition of diameter less than e. Furthermore, it is also easy to see that the map v w-~ h,,(T) on the set of T-invariant Borel probability measure M ( T ) is upper semi-continuous. It is a simple consequence of the convexity of the distance function that the geodesic flow on a compact manifold M of nonpositive curvature is entropy expansive. More precisely, one has: PROPOSITION 5.1. For each k E I~ one has that the time k-map ckk o f the geodesic flow on S M is entropy e~pansive. Choosing the metric dk on S M the number ~ = i n j ( M ) / 3 provides an expansivity constant f o r qbk.
Before we can sketch the proof of the uniqueness of the measure of maximal entropy we need to fix some notations. Let R be the constant introduced in Proposition 2.1 and p ~ .U be a fixed reference point in a fundamental domain f . If A C X we define S A to be the set of unit vectors with footpoint in A. For x E X and R > 0 let
D ( x , R' , R) -- {v E S B ( p ,
R
f
)lc,,(r)EB(x
,
R) f o r s o m e r > 0 ]
.
STEP 1. Let R' > 2R + 1/2 be a constant sati.~l~.,ing ~ C B ( p , R') and d ( p , X) >~ R' + R. Then we obtain
p ( O ( x , R' , R)) >~ c ~e -tul(p'x) f o r a constant c' > 0 not depending on X. This is a consequence of the local property of the measure # p . The next step follows from the convexity of the distance function. STEP 2. The cardinality o f (d,, 6)-separated set o f D ( x , R', R) is bounded from above by a constant a = a(6, R', R) f o r all x E X such that d ( x , p) ~ n + R' + R.
Hyperbolic dynamics and Riemannian geometry
533
Now consider a maximal 2R-separated set xl . . . . . xk(,,) of the geodesic sphere S ( p , r(n)) of radius r(n) = 2n + R' + R about p. Since this set is 2R-separated, the balls are pairwise disjoint and since it is maximal, the balls cover S B ( p , R'). Thus we can choose a partition F~" . . . . . F k(n) 2'' of S B ( p R') such that
D ( x i , R', R) C Fi21' C D ( x i , R', 2R) for each i. Let Q : S B ( p , R') --+ S M be the restriction to S B ( p , R') of the projection ~n S X --~ S M and let L 2n i - - Q ( F i- ) . Note that the map Q is at mostc~ to one for some constant u > 0. S T E P 3. L2n -- {,., 112n . . . . .
L2nk(,)} is a covering such that each v E S M lies in at most ot
sets from L 2'' 9 Furthermore, there is a constant [4 > 0 such that # ( L 2I" ) >~ f i e -2h'' f o r all i E {1 . . . . . k(n)}. The first assertion follows since Q is at most c~ to one. The second assertion follows from Step 1 and the fact that ~ ( A ) ~< o t ~ ( Q ( A ) ) for all measurable sets A C S B ( p , R'). To prove the uniqueness of the measure of maximal entropy we have to deal with partitions of small diameter. Let V = {vl . . . . . v,,,} be a maximal (d2,, ~)-separated set of S M . Choose a partition B such that for each B C / 3 there exists vi E V such that
B,12,, ( vi, ~/2) C B C B,i,,, (vi, ~). STEP 4. card{BE/31BNL
i211 =/=0]<~a(e.
, R t
,2R)
,
where a (s, R', 2R) the constant is the one defined in Step 2. Since
211
L i
--
Q( F i
3-.11
) we can lilt
to an e-separated set of F i2" C D ( x , R , 2R). Therefl)re, the result follows from Step 2. Now consider the pushforward A"
2n -- {qb"L i
} of the covering L2" by the time n map of
the geodesic flow. It follows from the definition of L 2'' that for each A E A" and any two vectors v, w E A there exists a continuous curve c~ :[0, 1 ] --~ S M connecting v and w such that the length of rr o 05tc~ is bounded by R' for all t E [--n, n]. The reason is that the set Fi2'' is contained in D ( x i , R ' , 2R). We will use the covering A" to separate measures in the following way. STEP 5. Let v, ~ be mutually singular Borel probability measures on S M such that the singular set has measure 0 with respect to I~. Then there exists a union C, o f subsets o f A" such that #(C,,) ~ O, v(C,,) ~ I f o r n ~ cx~.
534
G. Knieper
PROOF. Since # s v there exists a m e a s u r a b l e set I2 s.t. #(S2) = 0 and v(a'-2) = 1. By assumption we can assume that the singular set is contained in S-2. For each 3 > 0 one can choose c o m p a c t sets KI C a"-2 and K2 C SM\I-2 such that v ( K i ) ~> 1 - 3 and l z ( S M \ K 2 ) < 3. N o w define
Cm " - - U { A
6A'" IANKI
#dp }.
Then there exists n 6 1N such that for all v 6 K i, w 6 K2 and lifts fi, tb 6 S X d(rr4/fi, rrq~' tb) ~> R'
for some t 6 [ - n , n].
Otherwise one could prove, using a c o m p a c t n e s s argument and the flat strip T h e o r e m 1.5, the existence of a vector wo 6 K2 such that the geodesic c,,, 0 bounds a flat strip. This would contradict the fact that K2 is contained in the regular set. The set C,z covers K i by definition and C,, N K2 = ~b by the choice of n and the definition of A". Consequently, v(C,,) >~ 1 - 6 a n d / z ( C , , ) ~< l z ( S M \ K 2 ) <. 3. IS] The next standard estimate is used in the proof.
-- Z
ai l~ ai <~
i=1
ai
l
logk+-
i=1
(5.1)
e
provided a l . . . . . ak ~> 0 and Y~Y=I ai <~ 1. In the last step we will show that the measure of maximal entropy is unique. STEP 6. Let v be a Borel probability measure invariant under the geodesic flow. If v # then ht.(q5) < h = h,(49). PROOF. S i n c e / z is ergodic it is enough to prove that h~,(4~) < h if v and l~ are mutually singular. Fix e ~< i n j ( M ) / 3 and let V = {vl . . . . . v,,, } E S M be a maximal (e, d,, )-separated set. Choose a partition/3" such that for each B E/3" we have B,/~,, (vi, e / 2 ) C B C B,12,, (vi, ~) for some vi ~ V. Proposition 5.1 implies
2nh,,(ck') - h,,(ck2") = h , , ( c k 2 " , l Y ) <~ H,,(13") - - -
Z
v(B)logv(B).
B 613 ,t 1 2 n . . . . . . 1 2kn } of S M constructed between Step 2 and Step 3 Consider the covering L 2'' -- {'-'1
and its push forward {~b"L~"} = A". Since v and # are mutually singular and the singular set has measure 0 with respect to/~, Step 5 implies the existence of sets C,z consisting of a union of elements of A" such that # ( C , , ) ---, 0 and v(C,,) --+ 1 as n --~ cx~. Using the above estimate (5.1), we obtain: 2nh,(4~) ~< -
Z {1:IG13n Id)n Bfq(_'n#~}
v(B) log v(B) { B~13"
y~ 14~"BnC,, =r
l)(B) log v(B)
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536
G. Knieper
5.6. Growth rate o f singular closed geodesics In this last section we come back to the various counting functions of closed geodesics. Keeping the notation of Section 5.3 we denote by 79(M) a maximal set of primitive and geometrically distinct closed geodesics representing different free homotopy classes and by 7")reg(M) respectively 79sing(M) :-- 79(M) \ 7")reg(M) the subset of all regular respectively singular closed geodesics. Moreover, P(t), Preg(t) and esing(t) is the number of geodesics in 79(M), ~')reg(M) respectively ~sing(M) of length ~< t. It is a consequence of the uniqueness of the maximal measure, which we will denote by /Zmax that the topological entropy h(q~lsing ) of the geodesic flow restricted to the singular set is strictly smaller than the topological entropy h of the unrestricted flow. This can be seen by the following argument. The variational principle (see [57] or [83]) implies the existence of a sequence/z,, of invariant probability measures concentrated on the singular set such that
h(~lsing)/> hi,,, (4)) ~> h(qblsing) It is not difficult to prove that the map v --+ h,,(4~) defined on the space of 4~-invariant probability measures is upper semicontinuous with respect the weak topology [66]. Let/z be a weak limit of l~,,. Since the singular set is a closed subset of S M the limit measure is concentrated on the singular set as well. Therefore, the upper semi-continuity and the variational principle imply ht~ (4~) = h(q~lsing). On the other hand the maximal measure is concentrated on the regular set. Therefore, ~ # ~m,,x and the uniqueness of the measure of maximal entropy yields h > hj~ (4~) = h(q~lsing). As we will show, this implies that there are exponentially less singular than regular closed geodesics. THEOREM 6.1. There erists an e > 0 and t~) ~ 0 such that f o r all t >~ to P~ing(t) -~;t <~e Prcg(t) PROOF. Let a = lim sup I ---~,"x~
log P,;i ng (t) it
be the exponential growth rate of the singular closed geodesics. Since closed geodesics corresponding to different free homotopy classes are g-separated for some 6 > 0 it follows from the definition of the topological entropy that a ~< h(q~lsing). Hence, the theorem follows for any e > 0 smaller than h - h(q~lsing). I--I REMARK. For surfaces a stronger result holds. All orbits in the singular set have zero Lyapunov exponents and therefore the topological entropy of the singular set is zero. With the same argument as above one deduces that the growth rate of singular closed geodesics is subexponential. However, in higher dimensions graph manifolds provide rank 1 examples with an exponential growth rate of singular closed geodesics [66].
Hyperbolic dynamics and Riemannian geometry
537
As a last application, we obtain an estimate for the number of all closed geodesics. THEOREM 6.2. There exists a constant a > 1 such that f o r all t > 0 1 eh I < < . P ( t ) < ~a- e ht at t
PROOF. In Section 5.3 we derived such an estimate for the regular closed geodesics. Therefore, the lower bound is obvious and the upper bound follows from Theorem 6.1. [-1
6. Appendix 6.1. An intrinsic p r o o f o f Pestov's identity In this appendix we will give an intrinsic proof of a certain formula of Weitzenb6ck type on the tangent bundle of a Riemannian manifold. This formula, discovered by Pestov and Sharafutdinov [81], is called "Pestov's identity". The main difference to their formula is that it is stated in an invariant fashion avoiding the use of local coordinates. This identity is of particular interest for manifolds of nonpositive curvature and it should have many applications in dynamics and rigidity. For instance it has b,'en used by Croke and Sharafutdinov to show that compact manifolds of negative are is~,spectrically rigid, generalizing previous work of Guillemin and Kazhdan, and Min Oo (see Section 4.4). We will use the following notations and definitions. As usual let T M be the tangent bundle, S M the unit tangent bundle of a Riemannian manifold M, and 7r the canonical projection. Let ~0 6 C ~ ( T M ) be a infinitely often differentiable function and denote by grad~0(v) -- (grad h ~p(v), gradV q)(v)) 6 T , , T M the gradient of 99 with respect to the Sasaki metric (see Section I.I). The horizontal and vertical components grad h q)(v), grad v ~o(v) 6 T~rI,,IM are uniquely determined by the identities (grad" ~o(v), u,)"--
d 1=()
and {grad v 99(v), u,)"--
d
99(v + tu,) 1_--()
where for v, u, c T p M t --+ v,,,(t) denotes the parallel translation of v along the geodesic c,1, with initial conditions ~',,,(0) -u,. The horizontal and vertical gradient are examples of semibasic vector fields, i.e., smooth
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G. Knieper
542
Now observe that (grad v XGqg(v), grad h qg(v)} = {gradV{gradh qg(v), v), grad h qg(v))
--(Vgradh,p(v)grad h qg(v), v} + {grad h qg(v), grad h qg(v)}. Using this together with Step 1 and Step 2, we obtain Pestov's identity.
17
Now we are going to integrate Pestov's identity over the unit tangent bundle with respect to the Liouville measure dttr --d0p d vol(p), where d0p is the measure induced by the Riemannian metric on S p M and d vol is the associated volume element on M. LEMMA 1.2. Let X : T M --~ T M be a semibasic vector field. Then for each p ~ M divhX(v) d 0 p ( v ) = d i v ( L ' /, M
X(v)d0p(v))
/, M
and
,
div v X(v)dO/,(v)
/, M
L"I,M
+
- ,). f
I,M
REMARK. If X is homogeneous of degree m (i.e., X (iv) = i'" X (v) for i > 0), it follows that VVX(v) -- dl~=0X((l + t)v) -- m X ( v ) and consequently f div vX(v) d0p(v) - - ( n + m - l ) /
L, /, M
{X(v), v}dOp(v).
JS/, M
THEOREM 1.3. Let M be a compact n-dimensional Riemannian man~yold and q9 E C ~ ( T M). Then 2 L'M (gradh qg(v), grad v X(;qg(v))dpL
If q9 is homogeneous of degree m, we obtain
2 f'M {gradh ~o(v), grad v XGqg(v))dlzL
Hyperbolic dynamics and Riemannian geometry
- L" Ilgradh~~ M
543
+ (n + 2m) L M
LM
(R(grad v ~o(v),
v)v, grad v qg(v)) d/zL.
Acknowledgement I would like to thank Norbert Peyerimhoff and Karl Friedrich Siburg for for carefully reading this survey and useful discussions.
References Surveys in this volume [ 1] B. Hasselblatt, Hyperbolic dynamical systems, Handbook of Dynamical Systems, Vol. I A, B. Hasselblatt and A. Katok, eds, Elsevier, Amsterdam (2002), 239-319. 121 A. Katok and B. Hasselblatt, Principle structures, Handbook of Dynamical Systems, Vol. IA, B. Hasselblatt and A. Katok, eds, Elsevier, Amsterdam (2002), 1-203. 131 M. Pollicott, Periodic orbits and zetafimctions, Handbook of Dynamical Systems, Vol. I A, B. Hasselblatt and A. Katok, eds, Elsevier, Amsterdam (2(X)2), 409-453.
Other sources 14] D.V. Anosov, Geodesic flows on closed Riemannian man([ohls with negative curvature, Proc. Steklov. Inst. Math. 90 (1967). 151 W. Ballmann, Nonpositively curved man(fold~" of higher rank, Annals of Math. 122 (1985), 597-609. 161 W. Balhnann, Lectures on Spaces of Nonpositive Curvantre, DMV Seminar, Band 25 (1995). 171 W. Balhnann, M. Brin and P. Eberlein, Strt,'ture ofmanifohls ofnonpositive curvature, Annals of Math. 122 (1985), 171-203. 181 W. Ballmann, M. Gromov and V. Schroeder, Man(fold~" of Nonpositive Curvature, Progress in Math., Vol. 61, Birkhfiuser, Boston (1985). 191 Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov ?i distributions stable et instable d~ffdrentiables, J. Amer. Math. Soc. 5 (1992), 33-74. [ !()1 G. Besson, G. Courtois and S. Gailot, Entropies et rigiditds des espaces Iocah'ment synu;triques de courbure strictement ndgative, GAFA 5 (I 995), 731-799. [ 111 G. Besson, G. Courtois and S. Gallot, Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynamical Systems 16 (! 996), 623-649. [121 R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc. 164(1972), 321-331. 1131 R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1-30. [14] R. Bowen, Maximizing entropyfi~r a hyperbolic flow, Math. Systems Theory 7 (1973), 300-303. 1151 R. Bowen and B. Marcus, Unique ergodicity for horocvclefoliations, Israel J. Math. 36(1977), 43-67. 1161 M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grund. Math. Wissensch., Vol. 319, Springer (1999). [171 K. Bums, Private communication. [181 K. Burns and A. Katok, Manifi)lds with non-positive curvature, Ergodic Theory Dynamical Systems 5 ( ! 985), 307-317. 1191 K. Burns and R. Spatzier, Manifolds ofnonpositive curvature and their buildings, IHES 65 (1987), 35-59.
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