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On the Holonomy Groups of Linear Connections *)
MATHEMATICS
ON THE HOLONOMY GROUPS OF LINEAR CONNECTIONS *) II. PROPERTIES OF GENERAL LINEAR CONNECTIONS BY
ALBERT NIJENHUIS (Communicated by Prof. J. A. ScHOUTEN at the meeting of November 28, 1953)
§ 6. Linear connections in linear fibre bundles. In this section it is indicated how the concepts connected with a linear displacement in tangent spaces (affine connection), discussed in the first part of this paper 1 ), can be extended to general linear fibre bundles without any serious modification. First we define the concept of a linear fibre bundle 2 ) as we shall use it, and then that of a. linear connection in it. Let M be a manifold of class Ck; k): 1, and of dimension n; and associated with every point x EM let there be a linear N-dimensional real vector space F ~, called the fibre over x. The union B = u F x is mEM
called the bundle space, and the mapping n: B __,.. M, defined by n(a)=x if a E F "'' is called the projection map. Let there be a family ! of open sets covering M, and associated with every U E! a coordinate system (u) defined over U and a rectilinear coordinate system (A) for every F~, where x E U. If two open sets U, V of ! are the same as point sets in M, but the coordinate systems (u), (A) are not the same for both, then U and V are considered as distinct elements of _I. Consider U, U' E !, with corresponding coordinate systems (u), (A) and (u'), (A'); then at every x E U n U' there is a matrix with elements A~'(x)=(ox"'jox"), and also a matrix IIAf(x)ll belonging to the change of coordinates in Fx from (A) into (A'). If the family ! has the property that for all U, U' E! the coefficients Af(x) are functions of class C'; 0 l k over U n U', then the collection {B, M, n} is a linear fibre bundle of class Ck· 1• One easily shows that B is a manifold of class C 1, and that n is a mapping of the same class. Two sets _I, !' defining {B, M, n} as a fibre bundle are equivalent of class Qk.Z if the union ! U !' defines a fibre bundle of class Qk· 1• ! is
<<
*) Sponsored by the Office of Ordnance Research, U.S. Army, under Contract DA-36-034-0RD-1270. 1) Proc. Kon. Ned. Ak. v. Wet. Amsterdam A 56 (3), 233-249 (1953); Indagationes )!athematicae 15, (3), 233-249 (1953). 2) Our definition is a straightforward specialisation of N. STEENROD's; cf. The Topology of Fibre Bundles, (Princeton, 1951). See also C. EHRESMANN, Colloque de Topologie, (Brussels, 1950). 2 Series A
18
complete if 1:' C 1 whenever 1 and 1' are equivalent. Every set 1 has a unique completion that is equivalent to it. In what follows, 1 is assumed to be complete. Particular examples of linear fibre bundles are: (1) the tangent bundle: I!Afll= !lA~'! I; (2) the bundle of covariant vectors: IIA1'11= IIA:'II-1 ; (3) the bundle of E;s representing dh'(x) over a set Mii (all defined in § 5). The bundle {B, M, n} is called linearly connected, or provided with a linear displacement, of class em; m :'( l- 1 if over every U E 1 there is defined a set of nN2 real functions r,_.!i_(x), X E u of class em, in such a way that for every x E U n U'; U, U' E 1 one has (6. 1)
If uA is a vector field (local cross section) of class e1 over U, and if a curve in U defined by x=f(t); f of class e1 , then the expression
(6.2)
0 dt
U
A
duA
=dt
+
r
A
f.IB'U
B
e is
dxf.l dt
is defined at every x E e, and is a vector ofF.,. - The equation (bjdt)uA = 0 defines a mapping of the F"', x E e into each other, called the parallel displacement along e. Introducing matrix notations, equation (6.2) takes the form (2.4), and the explicit solution of (bjdt)u = 0 takes the form (2.5). With every curve e of class Dl, connecting points p, q E M there is thus associated a transformation mapping F:p into Fq. In particular, if p=q, e is a loop, and the corresponding mapping FP _,. FP is an element of the holonomy group f)(M, p) belonging to the connection r,_.~. The group f)(M, p) is the collection of the maps FP _,. FP belonging to all curves (of class D 1 ) beginning and ending at p. It is a Lie subgroup of the full group of linear transformations in N-dimensional vector space. The curves homotopic to zero lead to the restricted holonomy group f) 0 (M, p}, which is a normal subgroup of f)(M, p). The intersection n f) 0 (U, p) is called U=:Jp
the local holonomy group f)*(p). All this goes completely parallel to the correspqnding expositions in § 2. We also find again an infinitesimal holonomy group f)'(p), provided m=oo. It will be discussed in § 7. Following step by step the arguments in § 2, 3 one sees that only obvious changes have to be made in the formulations in order that they also apply to the general linear connections. Because the matrix notation is used many of the formulas retain exactly the same form, while the new interpretation is unambiguous. The equality of (2. 7) to the product integral in (2.5) also holds in the present case, as was shown in ScHLESINGER's paper. The curvature affinor R;~;A in (2. 7) (in full notation) is now defined by (6.3)
19
§ 7. The infinitesimal holonomy group for displacements in the fibres. The considerations of § 4 cannot immediately be extended to general fibre bundles (of class 0 00 ) because one cannot form covariant derivatives of the curvature affinor (6.3) unless one has, besides F"'j, also a connection 0
F"'). for the vectors and quantities of the tangent space. Introducing (7.la) or, in matrix notation, (7.lb) one sees by an argument analogous to that in the first part of § 4 that the matrices (7.2)
for every v• at x, are elements of df)*(p). The expression (7.la) is a quantity in the indices A, B; but not in v, fl, A.. In fact, the R;,~,;,;A are linear combinations of the R;~j/ and of the R;~~j/. Generally, we find a sequence of sets of matrices: (7.3)
defined by (7.4)
Under the change from the coordinate system (") to a system ("') the R.k' ... v,'p';.' are linear combinations of the R. 1...•,p;., 0 :::;; l :::;; k. Hence we have Lemma 7 .l. If U is a neighbourhood of p such that f)*(p) = 1) 0 ( U, p); if x is any point of U; and if T represents the mapping F.,-+ FP by parallel displacement along a curve in U, then the expressions (7.5)
TR'";. (x) T-I,
TR.'";. (x) T-I, ... ,TR.k····''";. (x) T-I, ...
all belong to df)*(p), and span df)*(p). The equivalent of Lemma 4.2 is now Lemma 7. 2. (7 .6)
The set of matrices R'";. (p), R.'";. (p), ... , R.k·····'";. (p), ...
spans a Lie algebra. Proof. (7. 7)
A direct computation gives 2 R[QG]vk···v,pJ.
=
[ RQG,
R.k···• p;.] • 1
If Rlkl denotes the space spanned by the matrices R.k····''""' then (7.7) shows: (7.8)
20
By repeated differentiation of (7. 7) one finds (7.9)
which proves the statement. Remark. The relation (7.9) is simpler than the corresponding (7.10), but the spaces RUe> now depend on the choice of the coordinate system (u). Only the spaces R(O> + R(l) + ... + R for each k are independent of the coordinate system (u). In order to show that Theorem 7 is valid in the present case we give the argument as it should be modified, starting after the justification of the statements (1) and (2) following Theorem 6. Since Rp,_(ta'-) and T(ta'-) are analytic functions of t for all points in U, the matrices of the set (7.10)
which together span d~*(p), are analytic functions of x, and admit a series expansion in powers of xa valid in U. The power series expansion has the form (7.11) )
Rp). (x) = Rp). (p)
= R~', (p)
p
+ taw Rwp). (p) + 2! aw•aw, Rw,w,p). (p) + ... + !
00
k=l
fk
kl awk ... aw, Rwk ... w,p). (p). .
This can be verified by using the procedure described in § 4 and by remarking that a~'(')Pa"= 0. Formula (7.11) shows that the Rp,_(x), which span d~*(p), belong to d~'(p), and this means that dim ~*(p)=dim ~'(p). Since p is arbitrary, Theorem 6 gives immediately that ~ 0 (M, p)=~'(p), and this proves Theorem 7 for the group ~o . . The last part that has to be modified slightly is the argument following the proof of Lemma 5.2. The formulas (5.6) are to be replaced by (7.12)
)
where O~~A and
B;;a o:~A,
R;;~A
=
R"wk ...
"AVttB -
W1
·ao·
B"wk···ro vp. 1
"A aB '
k = 1,2, ... ,
B;;a are connecting quantities, but the B for k > 0 are not. k
Indeed, the behaviour of the index a is that of a vector, but a linear combination of the one can derive
I
B~ 1 ... w,.~a; 0 :!'(
k
B~k' ... w,'•';,a
is
l :!'( k. Applying lle to (7.12)
(7.13)
from which follows the unique existence of coefficients (cf. (5.9)) (7.14)
rP~
such that
21 Again, one can show {
(7 .15)
[Oc, Ob] R;~~a =
c;;,a Oa, B;~c c;;,a =
while for the structure constants c;,;a(x) of the infinitesimal holonomy group ~'(x) the relations (5.13) are valid again. This is a short indication of how § 4, 5 have to be changed if there is a connection in the fibres only, and if· these fibres are not tangent spaces or spaces associated with them. It. may happen- and it will frequently happen in applications- that there is also a connection for the tangent
r ,:;., 0
spaces. The connection for the tangent bundle, say can then be used to define covariant derivatives of quantities that also have indices x, it, f-t, etc. We could give a recapituation of the whole treatment for that case, but the procedure will now be obvious. To give one typical example: the equivalent of (4.6) is now 0
(7.16) )
0
2 17rlll7a] v.k ...., RJ.I;. = - R;;,;kT 17T•k-1· .. •1 RJ.I;.- ... - R;;,;.T v.k ...., RJ.IT
+ [R(/G, 0
0
v.k ...., RJ.I;.] - 2 s;;,T
0
+
vT v.k ...., RJ.I"'
0
where R;~~" and S~~" belong to the connection rJ.<~. The commutator relation is again (4.10) instead of (7.8), and the R
17J.l cpA =""'cpA
+ FAj cpB =
0.
;
A, B = 0, ... , N,
where (8.3)
The F"'"Jj define a linear connection in the product bundle MxEN-'>-M, and to this connection there belong holonomy groups ~(M,p); ~ 0 (M,p); ~*(p) and ~'(p) for every pEM. Using the Theorems 9 and 10 one finds thus Theorem 12. The integrability conditions of (8.1) at a point p express the invariance of C/Ji(p) under the group ~'(p) of linear transformations. If II and 1JI are analytic these conditions are necessary and sufficient; i.e. to every cp•(p) invariant under q' (p) there is a solution of
22
(8.1) defined in a U(p) with tJ)i(p) as initial values at p. If II and lJI are of class 0 00 then the set of points p at which the invariance of t.Pi(p) under ~'(p) is sufficient for the existence of a local solution, is open and dense in M. § 9.
The non-homogeneous holonomy group.
Let M be an affinely
0
connected space with parameters r,_,:;.. Then there is, as E. CARTAN 3) remarked, besides the parallel displacement of tangent spaces, also a rolling or developing process for tangent spaces. Associated with a piecewise differentiable curve x(t), 0 ~ t ~ 1 there is now an affine mapping of the tangent space E~(x(O)) into E .. (x(1)}, which, in general, does not leave the origin £xed. The mapping is defined by the property that the tangent space E ..(x) is mapped into E ..(x+dx) by a parallel displacement followed by a translation over - dx. If u" are the components of a vector at x, and u"+du" those of its image at x+dx, then we have du"
Tt +
(9.1)
dx" r,_.A,. u .t dxf-1 Tt + Tt 0
=
o.
The image u"(t) at x(t) of a vector u"(O) at x(O) is given by (9.2)
where
u"(t) = THO,t) u.t(O) T~(O, t)
+ t"(O,t),
is the product integral in (2.5}, while t"(O, t) is given by
t" (0, t)
(9.3)
dx-l
t
f0 TA(T, t) -dT dT.
= -
The point t"(O, t) in E ..(x(t)) is t~ image of the origin of E,.(x(O)). Similarly ~"(t)=.rert"(t, 0) is the image of the origin of E .. (x(t)) in E,.(x(O)); i.e. ~"(t) is the image of the point x(t) itself in E,.(x(O}). Consequently the curve ~"(t); 0 t 1, is the image in E,.(x(O)) .of the curve x(t), 0 t 1 in M. By differentiation of
< <
< <
~" (t)
(~.4)
=
t
f0 T~ (t", 0)
d
dx
A
T
dt"
with respect to t one finds (9.5)
dk
f5k-l
dtk ~" (t) = TA(t, 0) dtk-1
dx.l
Tt
k > 0,
and consequently one has
<
Lemma 9.1. The kth derivative, 0 k, of the tangent vector of a curve at x(t), transported to x(O) by parallelism, is the same as the kth derivative of the tangent vector at ~(t) to the image of the given curve obtained by development into E .. (x(O)). If in any intrinsic way curvatures of a curve are defined (e.g. affine curvature, or curvature of curve in a Riemann space) one easily finds the well-known result 3)
Enseignement Mathematique 24, 1-18 (1924-25).
23
Lemma 9.2. A curve in M and its image in a tangent space under developments have the same curvatures at corresponding points. The existence theory of curves with given intrinsic equations provides the following converse of Lemma 9.2. Theorem 13. Whenever ~"(t), 0 ::S:; t ::S:; 1 is a curve of class D 1 in En(p), p EM and ~"(0) is the origin of En(p), there exists an s > 0 such that there is precisely one curve x(t), 0 ::S:; t ::S:; s in M, whose image under development into En(P) is the segment ~"(t), 0 ::S:; t ::S:; s. -If M is a complete Riemann space of positive definite metric, then s = l. called an affinely connected space complete if every curve ~(t), 0 ::S:; t ::S:; 1, of class D 1 in En(P); ~(0) = p, can wholly be developed into M.- We shall· use this concept of completeness later. The rolling process of the tangent space En(p), along all curves of class D 1 beginning and ending at p, gives rise to a group of linear non-homogeneous transformations of En(p), called the non-homogeneous holonomy group H(M,p) at p. One can again define the groups H 0 (M,p), H*(p) and H'(p). We give explicit expressions for dH'(p). In order to bring this nonhomogeneous group under the general scheme of § 6 we introduce in the tangent space an auxiliary coordinate u 0 which is kept equal to l. In terms of ui, i = 0, ... , n we have then for (9.1) EHRESMANN 4 )
(9.6)
where (9.7)
A direct computation of the curvature affinor (9.8)
R;~)." 0
= 0
where R;;;." and
0
R;~).", R;;~;,
S~)."
0
R;~;i
= 2 s;;x, R;;;_o =
gives
R;;~o
= 0,
belong to the given connection
0
r"1.
Covariant
0
derivation with rpl in the correction terms for all indices will be denoted 0
by V", whereas covariant differentiation with F"j in the correction terins for 0
indices i, j, ... and with rpl in the correction terms for indices x, A., ••• will be denoted by V" . - The relation between V"'k • V"'k- 1···"'• R;~ji and V"'k- 1···"'• R;~;i is found, using (9. 7): 0
... , V V R ..." V"'k • V"'k-1···"'• Rvp}. = "'k • "'k-1··· 001 vp}. ' 0
... , V · V R ... , V R ... " A" V R ...o Vwk • V. wk-1···"'• RvpO = wk • . wk_1 .o.w vpO - wk_ 1 ... w vpwk + wk wk-l···w •pO '
(9.9)
1
1
1
0
Vwk · Vwk_ 1... w 1 R"vp:A · ·o = Vwk · Vwk-1···"'1 R"vp}. · ·o 0
, 4)
Vwk · Vwk_ 1 ... w R···oVwk · Vwk_ 1 ... w R"··oVwk-l···w R"" ·0 vpO vpO vpwk • 1
1
Of. for instance op. cit.
2 ).
1
24 It follows now easily from (9.8, 9):
(9.10)
This proves : Theorem 14. The Lie algebra dH'(p) is spanned by the set of affinors 17"'k···"'' R;~ji ; k = 0, 1, 2, ... , whose explicit expressions are given by (9.10). We derive some further properties of the group H 0 (M, p). Theorem 15 5 ). If M is a manifold of symmetric affine connection, and if H'(p) is not trivial, then H 0 (M, p) has a non-trivial subgroup of affine rotations about the origin of E,..(p). 0
0
Proof. Let 17w 1... w, R;~)."=O at p for l=O, 1, ... , k-1, but not for l=k. Then the set 17"'k ... w, R;~ji has the property 17"'k""""'' R;~)." *- 0, but 17"'k···"'' R;~~" = 0. Hence 17"'k···"'' R;~ji generates at least one one-parameter subgroup of pure rotations. This is a subgroup of H'(p), and therefore it is a subgroup of H 0 (M, p). Theorem 16. H'(x) is not trivial EHRESMANN) then group of H 0 (M, p)
If M has a symmetric affine connection of 0 00 , if for any x EM, and if M is complete (in the sense of for every point 'YJ E E,..(p) there is a non-trivial subwhich leaves 'YJ invariant.
Proof. Develop the straight line P'YJ of E .. (p) into M, thus mapping 'YJ into a point y EM. Then H 0 (M, y) has a non-trivial subgroup which leaves y invariant. The injection of H 0 (M, y) into H 0 (M, p), by means of the geodesic segment py just constructed, maps the subgroup of H 0 (M, y) which leaves y invariant isomorphically onto the subgroup of H 0 (M, p) which leaves 'YJ invariant. This proves the theorem. Theorem l 7. If M is a complete Riemann space of positive definite metric, and if H 0 (M, p) leaves exactly one point invariant, then M is homeomorphic to Euclidean n-space. Proof. y
The construction in the proof of Theorem 16 gives the point
EM with the property that H 0 (M, y) consists of pure rotations. Every
point q E M can be connected with y by at least one geodesic. Hence the map q; : E,..(y) ~ M, established by developing all rays of E .. (y) into M, is onto whenever it is defined, and because M is complete it is defined for all points of E .. (y). The mapping q; is continuous. We complete the proof by showing that q; is a homeomorphism.- Suppose q; is not one-to-one, 5)
E.
CARTAN,
Acta Math. 48, l-42 (1926).
25 then there is a point q such that there are two geodesics yq. Then the image in En(Y) of the curve consisting of one geodesic yq and the other geodesic qy in succession has an endpoint r/ E En(Y) distinct from y. Injection of H 0(M, y) into itself by this curve then maps the invariant pointy into the point r/. Hence H 0 (M, y) has at least two invariant points, y and n'. This proves that there is no point q with two geodesics yq, and fP is one-to-one. -Let S'~~- 1 (r) be the solid open sphere of radius r in E .. (y). Then sn-1 (r) is compact. The mapping fP!S'~~-1 (r) into M is thus a continuous one-to-one mapping of a compact space into a Hausdorff space. This means that fP!S'~~-1 (r) is bicontinuous. Then fP- 1 is continuous at every point q of M, because a sufficiently small neighbourhood of q is contained in qJS'~~-1 (r) whenever r exceeds the geodesic distance between y and q. qJ is therefore a homeomorphism, and this completes the proof. Theorem 18 6 ). Whenever the holonomy group H 0 (M, p) of a space M with affine connection contains a one-parameter subgroup of translations, H 0 (M, p) contains either all translations, or h0 (M, p) is reducible. Proof. Denote a transformation of H 0 (M, p) by (T, t), where T is the "rotation" part, and t the translation part. Let (I,e r) be an infinitesimal pure translation of H 0 (M, p), and (T, t) any element of H 0 (M, p). Then (9.11)
(T, t) (I, e r) (T-1 ,
-
T-1 t)
=
(I, e Tr).
This shows not only that (I, e Tr) E H 0(M, p), and that the translations form a normal subgroup of H 0 (M, p), but since T E h0(M, p) it also shows that the vector space {r} spanned by the vectors r of all elements (I, e r) E H 0(M, p) is invariant under h0 (M, p). Unless this vector space spans the whole En(P) there is a non-trivial invariant subspace of h0 (M, p). This proves the theorem. · 1) When M is a Riemann space of positive definite metric the property that H 0 (M, p) contains no translations, is equivalent to the one that H 0 (M, p) has a fixed point. This (yet unpublished) theorem has been proved by S. SASAKI.
Institute for Advanced Study Princeton, N. J., U.S.A.
ON THE HOLONOMY GROUPS OF LINEAR CONNECTIONS *) II. PROPERTIES OF GENERAL LINEAR CONNECTIONS BY
ALBERT NIJENHUIS (Communicated by Prof. J. A. ScHOUTEN at the meeting of November 28, 1953)
§ 6. Linear connections in linear fibre bundles. In this section it is indicated how the concepts connected with a linear displacement in tangent spaces (affine connection), discussed in the first part of this paper 1 ), can be extended to general linear fibre bundles without any serious modification. First we define the concept of a linear fibre bundle 2 ) as we shall use it, and then that of a. linear connection in it. Let M be a manifold of class Ck; k): 1, and of dimension n; and associated with every point x EM let there be a linear N-dimensional real vector space F ~, called the fibre over x. The union B = u F x is mEM
called the bundle space, and the mapping n: B __,.. M, defined by n(a)=x if a E F "'' is called the projection map. Let there be a family ! of open sets covering M, and associated with every U E! a coordinate system (u) defined over U and a rectilinear coordinate system (A) for every F~, where x E U. If two open sets U, V of ! are the same as point sets in M, but the coordinate systems (u), (A) are not the same for both, then U and V are considered as distinct elements of _I. Consider U, U' E !, with corresponding coordinate systems (u), (A) and (u'), (A'); then at every x E U n U' there is a matrix with elements A~'(x)=(ox"'jox"), and also a matrix IIAf(x)ll belonging to the change of coordinates in Fx from (A) into (A'). If the family ! has the property that for all U, U' E! the coefficients Af(x) are functions of class C'; 0 l k over U n U', then the collection {B, M, n} is a linear fibre bundle of class Ck· 1• One easily shows that B is a manifold of class C 1, and that n is a mapping of the same class. Two sets _I, !' defining {B, M, n} as a fibre bundle are equivalent of class Qk.Z if the union ! U !' defines a fibre bundle of class Qk· 1• ! is
<<
*) Sponsored by the Office of Ordnance Research, U.S. Army, under Contract DA-36-034-0RD-1270. 1) Proc. Kon. Ned. Ak. v. Wet. Amsterdam A 56 (3), 233-249 (1953); Indagationes )!athematicae 15, (3), 233-249 (1953). 2) Our definition is a straightforward specialisation of N. STEENROD's; cf. The Topology of Fibre Bundles, (Princeton, 1951). See also C. EHRESMANN, Colloque de Topologie, (Brussels, 1950). 2 Series A
18
complete if 1:' C 1 whenever 1 and 1' are equivalent. Every set 1 has a unique completion that is equivalent to it. In what follows, 1 is assumed to be complete. Particular examples of linear fibre bundles are: (1) the tangent bundle: I!Afll= !lA~'! I; (2) the bundle of covariant vectors: IIA1'11= IIA:'II-1 ; (3) the bundle of E;s representing dh'(x) over a set Mii (all defined in § 5). The bundle {B, M, n} is called linearly connected, or provided with a linear displacement, of class em; m :'( l- 1 if over every U E 1 there is defined a set of nN2 real functions r,_.!i_(x), X E u of class em, in such a way that for every x E U n U'; U, U' E 1 one has (6. 1)
If uA is a vector field (local cross section) of class e1 over U, and if a curve in U defined by x=f(t); f of class e1 , then the expression
(6.2)
0 dt
U
A
duA
=dt
+
r
A
f.IB'U
B
e is
dxf.l dt
is defined at every x E e, and is a vector ofF.,. - The equation (bjdt)uA = 0 defines a mapping of the F"', x E e into each other, called the parallel displacement along e. Introducing matrix notations, equation (6.2) takes the form (2.4), and the explicit solution of (bjdt)u = 0 takes the form (2.5). With every curve e of class Dl, connecting points p, q E M there is thus associated a transformation mapping F:p into Fq. In particular, if p=q, e is a loop, and the corresponding mapping FP _,. FP is an element of the holonomy group f)(M, p) belonging to the connection r,_.~. The group f)(M, p) is the collection of the maps FP _,. FP belonging to all curves (of class D 1 ) beginning and ending at p. It is a Lie subgroup of the full group of linear transformations in N-dimensional vector space. The curves homotopic to zero lead to the restricted holonomy group f) 0 (M, p}, which is a normal subgroup of f)(M, p). The intersection n f) 0 (U, p) is called U=:Jp
the local holonomy group f)*(p). All this goes completely parallel to the correspqnding expositions in § 2. We also find again an infinitesimal holonomy group f)'(p), provided m=oo. It will be discussed in § 7. Following step by step the arguments in § 2, 3 one sees that only obvious changes have to be made in the formulations in order that they also apply to the general linear connections. Because the matrix notation is used many of the formulas retain exactly the same form, while the new interpretation is unambiguous. The equality of (2. 7) to the product integral in (2.5) also holds in the present case, as was shown in ScHLESINGER's paper. The curvature affinor R;~;A in (2. 7) (in full notation) is now defined by (6.3)
19
§ 7. The infinitesimal holonomy group for displacements in the fibres. The considerations of § 4 cannot immediately be extended to general fibre bundles (of class 0 00 ) because one cannot form covariant derivatives of the curvature affinor (6.3) unless one has, besides F"'j, also a connection 0
F"'). for the vectors and quantities of the tangent space. Introducing (7.la) or, in matrix notation, (7.lb) one sees by an argument analogous to that in the first part of § 4 that the matrices (7.2)
for every v• at x, are elements of df)*(p). The expression (7.la) is a quantity in the indices A, B; but not in v, fl, A.. In fact, the R;,~,;,;A are linear combinations of the R;~j/ and of the R;~~j/. Generally, we find a sequence of sets of matrices: (7.3)
defined by (7.4)
Under the change from the coordinate system (") to a system ("') the R.k' ... v,'p';.' are linear combinations of the R. 1...•,p;., 0 :::;; l :::;; k. Hence we have Lemma 7 .l. If U is a neighbourhood of p such that f)*(p) = 1) 0 ( U, p); if x is any point of U; and if T represents the mapping F.,-+ FP by parallel displacement along a curve in U, then the expressions (7.5)
TR'";. (x) T-I,
TR.'";. (x) T-I, ... ,TR.k····''";. (x) T-I, ...
all belong to df)*(p), and span df)*(p). The equivalent of Lemma 4.2 is now Lemma 7. 2. (7 .6)
The set of matrices R'";. (p), R.'";. (p), ... , R.k·····'";. (p), ...
spans a Lie algebra. Proof. (7. 7)
A direct computation gives 2 R[QG]vk···v,pJ.
=
[ RQG,
R.k···• p;.] • 1
If Rlkl denotes the space spanned by the matrices R.k····''""' then (7.7) shows: (7.8)
20
By repeated differentiation of (7. 7) one finds (7.9)
which proves the statement. Remark. The relation (7.9) is simpler than the corresponding (7.10), but the spaces RUe> now depend on the choice of the coordinate system (u). Only the spaces R(O> + R(l) + ... + R
which together span d~*(p), are analytic functions of x, and admit a series expansion in powers of xa valid in U. The power series expansion has the form (7.11) )
Rp). (x) = Rp). (p)
= R~', (p)
p
+ taw Rwp). (p) + 2! aw•aw, Rw,w,p). (p) + ... + !
00
k=l
fk
kl awk ... aw, Rwk ... w,p). (p). .
This can be verified by using the procedure described in § 4 and by remarking that a~'(')Pa"= 0. Formula (7.11) shows that the Rp,_(x), which span d~*(p), belong to d~'(p), and this means that dim ~*(p)=dim ~'(p). Since p is arbitrary, Theorem 6 gives immediately that ~ 0 (M, p)=~'(p), and this proves Theorem 7 for the group ~o . . The last part that has to be modified slightly is the argument following the proof of Lemma 5.2. The formulas (5.6) are to be replaced by (7.12)
)
where O~~A and
B;;a o:~A,
R;;~A
=
R"wk ...
"AVttB -
W1
·ao·
B"wk···ro vp. 1
"A aB '
k = 1,2, ... ,
B;;a are connecting quantities, but the B for k > 0 are not. k
Indeed, the behaviour of the index a is that of a vector, but a linear combination of the one can derive
I
B~ 1 ... w,.~a; 0 :!'(
k
B~k' ... w,'•';,a
is
l :!'( k. Applying lle to (7.12)
(7.13)
from which follows the unique existence of coefficients (cf. (5.9)) (7.14)
rP~
such that
21 Again, one can show {
(7 .15)
[Oc, Ob] R;~~a =
c;;,a Oa, B;~c c;;,a =
while for the structure constants c;,;a(x) of the infinitesimal holonomy group ~'(x) the relations (5.13) are valid again. This is a short indication of how § 4, 5 have to be changed if there is a connection in the fibres only, and if· these fibres are not tangent spaces or spaces associated with them. It. may happen- and it will frequently happen in applications- that there is also a connection for the tangent
r ,:;., 0
spaces. The connection for the tangent bundle, say can then be used to define covariant derivatives of quantities that also have indices x, it, f-t, etc. We could give a recapituation of the whole treatment for that case, but the procedure will now be obvious. To give one typical example: the equivalent of (4.6) is now 0
(7.16) )
0
2 17rlll7a] v.k ...., RJ.I;. = - R;;,;kT 17T•k-1· .. •1 RJ.I;.- ... - R;;,;.T v.k ...., RJ.IT
+ [R(/G, 0
0
v.k ...., RJ.I;.] - 2 s;;,T
0
+
vT v.k ...., RJ.I"'
0
where R;~~" and S~~" belong to the connection rJ.<~. The commutator relation is again (4.10) instead of (7.8), and the R
17J.l cpA =""'cpA
+ FAj cpB =
0.
;
A, B = 0, ... , N,
where (8.3)
The F"'"Jj define a linear connection in the product bundle MxEN-'>-M, and to this connection there belong holonomy groups ~(M,p); ~ 0 (M,p); ~*(p) and ~'(p) for every pEM. Using the Theorems 9 and 10 one finds thus Theorem 12. The integrability conditions of (8.1) at a point p express the invariance of C/Ji(p) under the group ~'(p) of linear transformations. If II and 1JI are analytic these conditions are necessary and sufficient; i.e. to every cp•(p) invariant under q' (p) there is a solution of
22
(8.1) defined in a U(p) with tJ)i(p) as initial values at p. If II and lJI are of class 0 00 then the set of points p at which the invariance of t.Pi(p) under ~'(p) is sufficient for the existence of a local solution, is open and dense in M. § 9.
The non-homogeneous holonomy group.
Let M be an affinely
0
connected space with parameters r,_,:;.. Then there is, as E. CARTAN 3) remarked, besides the parallel displacement of tangent spaces, also a rolling or developing process for tangent spaces. Associated with a piecewise differentiable curve x(t), 0 ~ t ~ 1 there is now an affine mapping of the tangent space E~(x(O)) into E .. (x(1)}, which, in general, does not leave the origin £xed. The mapping is defined by the property that the tangent space E ..(x) is mapped into E ..(x+dx) by a parallel displacement followed by a translation over - dx. If u" are the components of a vector at x, and u"+du" those of its image at x+dx, then we have du"
Tt +
(9.1)
dx" r,_.A,. u .t dxf-1 Tt + Tt 0
=
o.
The image u"(t) at x(t) of a vector u"(O) at x(O) is given by (9.2)
where
u"(t) = THO,t) u.t(O) T~(O, t)
+ t"(O,t),
is the product integral in (2.5}, while t"(O, t) is given by
t" (0, t)
(9.3)
dx-l
t
f0 TA(T, t) -dT dT.
= -
The point t"(O, t) in E ..(x(t)) is t~ image of the origin of E,.(x(O)). Similarly ~"(t)=.rert"(t, 0) is the image of the origin of E .. (x(t)) in E,.(x(O)); i.e. ~"(t) is the image of the point x(t) itself in E,.(x(O}). Consequently the curve ~"(t); 0 t 1, is the image in E,.(x(O)) .of the curve x(t), 0 t 1 in M. By differentiation of
< <
< <
~" (t)
(~.4)
=
t
f0 T~ (t", 0)
d
dx
A
T
dt"
with respect to t one finds (9.5)
dk
f5k-l
dtk ~" (t) = TA(t, 0) dtk-1
dx.l
Tt
k > 0,
and consequently one has
<
Lemma 9.1. The kth derivative, 0 k, of the tangent vector of a curve at x(t), transported to x(O) by parallelism, is the same as the kth derivative of the tangent vector at ~(t) to the image of the given curve obtained by development into E .. (x(O)). If in any intrinsic way curvatures of a curve are defined (e.g. affine curvature, or curvature of curve in a Riemann space) one easily finds the well-known result 3)
Enseignement Mathematique 24, 1-18 (1924-25).
23
Lemma 9.2. A curve in M and its image in a tangent space under developments have the same curvatures at corresponding points. The existence theory of curves with given intrinsic equations provides the following converse of Lemma 9.2. Theorem 13. Whenever ~"(t), 0 ::S:; t ::S:; 1 is a curve of class D 1 in En(p), p EM and ~"(0) is the origin of En(p), there exists an s > 0 such that there is precisely one curve x(t), 0 ::S:; t ::S:; s in M, whose image under development into En(P) is the segment ~"(t), 0 ::S:; t ::S:; s. -If M is a complete Riemann space of positive definite metric, then s = l. called an affinely connected space complete if every curve ~(t), 0 ::S:; t ::S:; 1, of class D 1 in En(P); ~(0) = p, can wholly be developed into M.- We shall· use this concept of completeness later. The rolling process of the tangent space En(p), along all curves of class D 1 beginning and ending at p, gives rise to a group of linear non-homogeneous transformations of En(p), called the non-homogeneous holonomy group H(M,p) at p. One can again define the groups H 0 (M,p), H*(p) and H'(p). We give explicit expressions for dH'(p). In order to bring this nonhomogeneous group under the general scheme of § 6 we introduce in the tangent space an auxiliary coordinate u 0 which is kept equal to l. In terms of ui, i = 0, ... , n we have then for (9.1) EHRESMANN 4 )
(9.6)
where (9.7)
A direct computation of the curvature affinor (9.8)
R;~)." 0
= 0
where R;;;." and
0
R;~).", R;;~;,
S~)."
0
R;~;i
= 2 s;;x, R;;;_o =
gives
R;;~o
= 0,
belong to the given connection
0
r"1.
Covariant
0
derivation with rpl in the correction terms for all indices will be denoted 0
by V", whereas covariant differentiation with F"j in the correction terins for 0
indices i, j, ... and with rpl in the correction terms for indices x, A., ••• will be denoted by V" . - The relation between V"'k • V"'k- 1···"'• R;~ji and V"'k- 1···"'• R;~;i is found, using (9. 7): 0
... , V V R ..." V"'k • V"'k-1···"'• Rvp}. = "'k • "'k-1··· 001 vp}. ' 0
... , V · V R ... , V R ... " A" V R ...o Vwk • V. wk-1···"'• RvpO = wk • . wk_1 .o.w vpO - wk_ 1 ... w vpwk + wk wk-l···w •pO '
(9.9)
1
1
1
0
Vwk · Vwk_ 1... w 1 R"vp:A · ·o = Vwk · Vwk-1···"'1 R"vp}. · ·o 0
, 4)
Vwk · Vwk_ 1 ... w R···oVwk · Vwk_ 1 ... w R"··oVwk-l···w R"" ·0 vpO vpO vpwk • 1
1
Of. for instance op. cit.
2 ).
1
24 It follows now easily from (9.8, 9):
(9.10)
This proves : Theorem 14. The Lie algebra dH'(p) is spanned by the set of affinors 17"'k···"'' R;~ji ; k = 0, 1, 2, ... , whose explicit expressions are given by (9.10). We derive some further properties of the group H 0 (M, p). Theorem 15 5 ). If M is a manifold of symmetric affine connection, and if H'(p) is not trivial, then H 0 (M, p) has a non-trivial subgroup of affine rotations about the origin of E,..(p). 0
0
Proof. Let 17w 1... w, R;~)."=O at p for l=O, 1, ... , k-1, but not for l=k. Then the set 17"'k ... w, R;~ji has the property 17"'k""""'' R;~)." *- 0, but 17"'k···"'' R;~~" = 0. Hence 17"'k···"'' R;~ji generates at least one one-parameter subgroup of pure rotations. This is a subgroup of H'(p), and therefore it is a subgroup of H 0 (M, p). Theorem 16. H'(x) is not trivial EHRESMANN) then group of H 0 (M, p)
If M has a symmetric affine connection of 0 00 , if for any x EM, and if M is complete (in the sense of for every point 'YJ E E,..(p) there is a non-trivial subwhich leaves 'YJ invariant.
Proof. Develop the straight line P'YJ of E .. (p) into M, thus mapping 'YJ into a point y EM. Then H 0 (M, y) has a non-trivial subgroup which leaves y invariant. The injection of H 0 (M, y) into H 0 (M, p), by means of the geodesic segment py just constructed, maps the subgroup of H 0 (M, y) which leaves y invariant isomorphically onto the subgroup of H 0 (M, p) which leaves 'YJ invariant. This proves the theorem. Theorem l 7. If M is a complete Riemann space of positive definite metric, and if H 0 (M, p) leaves exactly one point invariant, then M is homeomorphic to Euclidean n-space. Proof. y
The construction in the proof of Theorem 16 gives the point
EM with the property that H 0 (M, y) consists of pure rotations. Every
point q E M can be connected with y by at least one geodesic. Hence the map q; : E,..(y) ~ M, established by developing all rays of E .. (y) into M, is onto whenever it is defined, and because M is complete it is defined for all points of E .. (y). The mapping q; is continuous. We complete the proof by showing that q; is a homeomorphism.- Suppose q; is not one-to-one, 5)
E.
CARTAN,
Acta Math. 48, l-42 (1926).
25 then there is a point q such that there are two geodesics yq. Then the image in En(Y) of the curve consisting of one geodesic yq and the other geodesic qy in succession has an endpoint r/ E En(Y) distinct from y. Injection of H 0(M, y) into itself by this curve then maps the invariant pointy into the point r/. Hence H 0 (M, y) has at least two invariant points, y and n'. This proves that there is no point q with two geodesics yq, and fP is one-to-one. -Let S'~~- 1 (r) be the solid open sphere of radius r in E .. (y). Then sn-1 (r) is compact. The mapping fP!S'~~-1 (r) into M is thus a continuous one-to-one mapping of a compact space into a Hausdorff space. This means that fP!S'~~-1 (r) is bicontinuous. Then fP- 1 is continuous at every point q of M, because a sufficiently small neighbourhood of q is contained in qJS'~~-1 (r) whenever r exceeds the geodesic distance between y and q. qJ is therefore a homeomorphism, and this completes the proof. Theorem 18 6 ). Whenever the holonomy group H 0 (M, p) of a space M with affine connection contains a one-parameter subgroup of translations, H 0 (M, p) contains either all translations, or h0 (M, p) is reducible. Proof. Denote a transformation of H 0 (M, p) by (T, t), where T is the "rotation" part, and t the translation part. Let (I,e r) be an infinitesimal pure translation of H 0 (M, p), and (T, t) any element of H 0 (M, p). Then (9.11)
(T, t) (I, e r) (T-1 ,
-
T-1 t)
=
(I, e Tr).
This shows not only that (I, e Tr) E H 0(M, p), and that the translations form a normal subgroup of H 0 (M, p), but since T E h0(M, p) it also shows that the vector space {r} spanned by the vectors r of all elements (I, e r) E H 0(M, p) is invariant under h0 (M, p). Unless this vector space spans the whole En(P) there is a non-trivial invariant subspace of h0 (M, p). This proves the theorem. · 1) When M is a Riemann space of positive definite metric the property that H 0 (M, p) contains no translations, is equivalent to the one that H 0 (M, p) has a fixed point. This (yet unpublished) theorem has been proved by S. SASAKI.
Institute for Advanced Study Princeton, N. J., U.S.A.