A Cheeger finiteness theorem for Finsler manifolds

J. Math. Anal. Appl. 433 (2016) 1690–1717

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

A Cheeger finiteness theorem for Finsler manifolds Wei Zhao a,∗ , Yibing Shen b a

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China b School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China

a r t i c l e

i n f o

Article history: Received 17 May 2015 Available online 1 September 2015 Submitted by H.R. Parks

a b s t r a c t In this paper, we establish a Cheeger finiteness theorem for Berwald manifolds. Moreover, a Cheeger type estimate on injectivity radii for Finsler manifolds is obtained and the existence of the center of mass of a Berwald manifold is proved. © 2015 Elsevier Inc. All rights reserved.

Keywords: Finsler manifold Finiteness theorem Injectivity radius Center of mass

1. Introduction Finiteness theorems are theorems giving bounds on certain geometrical quantities such that the family of manifolds admitting metrics which satisfy the bounds is finite up to homotopy equivalence, or homeomorphism, of diffeomorphism. A famous finiteness result in Riemannian geometry is Cheeger’s finiteness theorem [7,10,14], which says that given n ∈ N and k, D, V > 0, there are only finitely many diffeomorphism types among compact Riemannian n-manifolds (M n , g) satisfying |KM | ≤ k, diam(M ) ≤ D and Vol(M ) ≥ V . Note that according to [10] and [7, Theorem 2.1], the assumptions of Cheeger’s theorem eliminate the collapsing case. Also refer to [1,6,15] for more details. Finsler metrics are just Riemannian metrics without quadratic restriction. It is a natural problem that whether an analogue of Cheeger’s theorem still holds in the Finslerian case. A Finsler metric F on a manifold M is called Berwaldian if there is a linear Berwald connection such that all the tangent spaces of M are linearly isometric to a common Minkowski space. Berwald metrics are a kind of important Finsler metrics, which are more general than Riemannian and locally Minkowskian metrics and therefore can be viewed as a median between Riemannian metrics and general Finsler metrics (cf. [22,3]). It is also noticeable that there are infinitely many Berwald metrics which are neither Riemannian nor locally Minkowskian. * Corresponding author. E-mail addresses: [email protected] (W. Zhao), [email protected] (Y. Shen). http://dx.doi.org/10.1016/j.jmaa.2015.08.067 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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Example 1. (See [3].) Set M = S2 × S1 . Let α be the canonical Riemannian product metric on M and let t be the usual spherical coordinate of S. Then F := α +  dt, for  ∈ (0, 1), is a family of Bewald(–Randers) metrics globally defined on M with non-negative flag curvature. Example 2. (See [8].) Given an arbitrary smooth function f : [0, ∞) × [0, ∞) → [0, ∞) with f (λs, λt) = λf (s, t) for all λ > 0, let (Mi , gi ), i = 1, 2 be two arbitrary Riemannian manifolds. Then F (x, y) :=

 f [g1 (x1 , y1 ), g2 (x2 , y2 )]

is a Berwald metric on M1 ×M2 , where x = (x1 , x2 ) ∈ M and y = y1 ⊕y2 ∈ Tx (M1 ×M2 ) ∼ = Tx1 M1 ⊕Tx2 M2 . More generally, Z. Szabó in [22,23] points out that any Berwald metric is a perturbed-Cartesian product of Riemannian, Minkowski and such non-Riemannian metrics which can be constructed on irreducible symmetric manifolds of rank > 1. Refer to [3,8,19,22,23,27] for more interesting examples. We also remark that there is a long existing problem concerning Berwald metrics, that is, whether or not any Landsberg metric is a Berwald metric. Recently, Ben–Shen [4], Z. Shen [20] and Z. Szabó [24,25] have given affirmative answers under additional assumptions, but the problem is still open in the general case. Hence, the class of Berwald metrics is a large and interesting class, and the study on Berwald manifolds will enhance our understanding of Finsler geometry. The main purpose of this paper is to study the finiteness problem above in the Berwaldian case. In [22], Z. Szabó finds an important relationship between Berwald metrics and Riemannian metrics, that is, there always exists a Riemannian metric such that its Levi-Civita connection coincides the Chern connection of a given Berwald manifold. For this reason, the curvature properties of a Berwald metric are similar to ones of a Riemannian metric (cf. [3,19]). Even so, geometric-topological properties of Berwald manifolds are much different from those of Riemannian manifolds. For example, a compact manifold M with a family of Riemannian metrics {gm } such that all (M, gm ) satisfy the assumptions of Cheeger’s theorem cannot collapse. However, this is not true for Berwald metrics. Example 3. Define a sequence of Berwald metrics on T2 = S1 × S1 by 

Fm

1 := α + 1 − m

 β, m ≥ 1,

where α is the canonical Riemannian product metric on T2 , and β is a parallel 1-form on T2 with β α = 1. Then {(T2 , Fm )}m satisfy √ Km = 0, diamm (T2 ) ≤ 2( 2 + 1)π, μm (T2 ) = 4π 2 , where μm is the Holmes–Thompson volume of (T2 , Fm ). However, the injectivity radius im (T2 ) → 0 as m → +∞. Before explaining the reason of the collapsing above, we introduce the uniformity constant ΛF of a Finsler manifold (M, F ), which is a non-Riemannian geometric quantity (cf. [9]). The uniformity constant is a scalar, which is defined by ΛF :=

sup X,Y,Z∈SM

gX (Y, Y ) , gZ (Y, Y )

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where g is the fundamental tensor of F . Clearly, ΛF ≥ 1 with equality if and only if F is Riemannian. Roughly speaking, the injectivity radius is inversely proportional to the uniformity constant. And a direct calculation yields that the uniformity constants Λm in Example 1 satisfy Λm ≥ (2m − 1)2 → ∞, as m → ∞, which makes (T2 , Fm ) collapse. On the other hand, if a compact n-dimensional manifold with a family of Berwald metrics {Fm } satisfies Λm ≤ Λ, |Km | ≤ k, diamm (M ) ≤ D and μm (M ) ≥ V , then (M, Fm ) cannot collapse. In fact, we shall establish the following Cheeger type estimate on the injectivity radius   1 (n − 1)V 1 (1 + Λ− 2 )π √ , . im ≥ 1 min 3n 1 + Λ2 k cn−2 Λ 2 sn−1 −k (D) See Theorem 3.4 below for an estimate on general Finsler manifolds. Inspired by the observation above, we investigate the diffeomorphism types of compact Berwald n-manifolds with a uniform upper bound on the uniformity constants and the assumptions of Cheeger’s theorem. Then we shall establish the following Theorem 1.1. Given n ∈ N, Λ ≥ 1, k ≥ 0 and V, D > 0, there exist only finitely many diffeomorphism classes of compact Berwald n-manifolds (M, F ) satisfying ΛF ≤ Λ, |KM | ≤ k, μ(M ) ≥ V, diam(M ) ≤ D, where μ(M ) is either the Busemann–Hausdorff volume or the Holmes–Thompson volume of M . It is easy to see that Theorem 1.1 implies the original Cheeger’s theorem. Our main tool is a generalized Peter’s lemma (see Section 6 below). It should be remarked that with additional assumptions on T-curvature and Riemannian curvature, our method works for general Finsler metrics. This will be discussed somewhere else. We also remark that Gromov in [10] obtains Cheeger’s theorem by a remarkable theorem, which is called Gromov’s convergence theorem in [12]. However, this theorem cannot be extended to the Finsler setting (even to the Berwald setting), because it requires the uniformity constant to be almost equal to 1. Refer to [19,21,26,28] for other finiteness theorems in Finsler geometry. The arrangement of contents of this paper is as follows. In Section 2, we brief some necessary definitions and properties concerned with Finsler geometry. In Section 3, the Finslerian versions of Klingenberg’s theorem and Cheeger’s estimate on injectivity radii are established. In Section 4, we estimate the convex radius and study the center of mass of a Berwald manifold. Theorem 1.1 is proved in Section 5 by a generalized Peter’s lemma, and the latter is proved in Section 6. In Appendix A, we give some estimates for Jacobi fields on Finsler manifolds. In Appendix B, we study the parallel transformations on a Berwald manifold. 2. Preliminaries In this section, we recall some definitions and properties about Finsler manifolds. See [3,19] for more details. Let (M, F ) be a (connected) Finsler n-manifold with Finsler metric F : T M → [0, ∞). Define Sx M := {y ∈ Tx M : F (x, y) = 1} and SM := ∪x∈M Sx M . Let (x, y) = (xi , y i ) be local coordinates on T M . Define yi 1 ∂ 2 F 2 (x, y) , gij (x, y) := , F 2 ∂y i ∂y j   ∂gjl 1 ∂gkl ∂gjk , := g il + − 2 ∂xk ∂xj ∂xl

i := i γjk

F ∂ 3 F 2 (x, y) , 4 ∂y i ∂y j ∂y k  i j  k r s Nji := γjk  − Aijk γrs   · F. Aijk (x, y) :=

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The Chern connection ∇ is defined on the pulled-back bundle π ∗ T M and its forms are characterized by the following structure equations: (1) Torsion freeness: dxj ∧ ωji = 0; A (2) Almost g-compatibility: dgij − gkj ωik − gik ωjk = 2 Fijk (dy k + Nlk dxl ). From above, it’s easy to obtain ωji = Γijk dxk , and Γijk = Γikj . It should be remarked that Γikj = Γikj (x, y) is a local smooth function on SM . In particular, F is called a Berwald metric if ∂Γikj /∂y s = 0. The curvature form of the Chern connection is defined as Ωij := dωji − ωjk ∧ ωki =:

1 i dy l + Nsl dxs Rj kl dxk ∧ dxl + Pji kl dxk ∧ . 2 F

Given a non-zero vector V ∈ Tx M , the flag curvature K(y, V ) on (x, y) ∈ T M \0 is defined as K(y, V ) :=

V i y j Rjikl y l V k , gy (y, y)gy (V, V ) − [gy (y, V )]2

where Rjikl := gis Rjs kl . The reversibility λF and the uniformity constant ΛF of (M, F ) are defined as λF := sup X∈SM

F (−X) gX (Y, Y ) , ΛF := . sup F (X) X,Y,Z∈SM gZ (Y, Y )

Clearly, λF ≥ 1 with equality if and only if F is reversible, and ΛF ≥ 1 with equality if and only if F is √ Riemannian. In particular, λF ≤ ΛF . The Legendre transformation L : T M → T ∗ M is defined by  L(Y ) =

0, Y = 0,  0. gY (Y, ·), Y =

For each x ∈ M , the Legendre transformation is a smooth diffeomorphism from Tx M \{0} to Tx∗ M \{0}. The average Riemannian metric g˜ induced by F is defined by 1 g˜(X, Y ) = Vol(x)

gy (X, Y )dνx (y), ∀ X, Y ∈ Tx M, Sx M

where Vol(x) =

Sx M

dνx (y), and dνx is the Riemannian volume form of Sx M induced by F .

3. The injectivity radius of a Finsler manifold Let (M, F ) be a compact Finsler manifold. There exist two points p and q such that iM = d(p, q) = d(p, Cutp ). Let γy (t), 0 ≤ t ≤ d(p, q) be a normal minimal geodesic from p to q. Then q is the cut point of p along γy . If q is not the first conjugate point of p along γy , [3, Proposition 8.2.1] implies that there exists another distinct normal minimal geodesic γw (t), 0 ≤ t ≤ d(p, q) from p to q. If F is reversible, [19, Lemma 12.2.5] yields γ˙ y (d(p, q)) = −γ˙ w (d(p, q)) and hence, Klingenberg’s theorem [13] still holds in the reversible Finsler case. If F is nonreversible, then the first variation formula [3, p. 123] together with [17, Lemma 9.3] yields

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that Lγ˙ y (d(p,q)) = −Lγ˙ w (d(p,q)) . However, one cannot deduce γ˙ y (d(p, q)) = −γ˙ w (d(p, q)), since the Legendre transformation is non-linear. Hence, it is an interesting and important question that whether Klingenberg’s theorem holds in the nonreversible case. In the following, we use the methods of Rademacher in [16–18] to establish a Finslerian version of Klingenberg’s theorem. Theorem 3.1. Let (M, F ) be a compact Finsler manifold with KM ≤ k. Then  iM ≥ min

1 π √ , the shortest simple closed geodesic in M λF k 1 + λF

.

In particular, the equality holds if F is reversible (i.e., λF = 1). Proof. As in [16,17], set ˜ q) : q ∈ Cutp }, ˜iM := inf ˜ip , ˜ip := inf{d(p, p∈M

˜ q) := 1 (d(p, q) + d(q, p)). Clearly, where d(p, 2 1 + λ−1 (1 + λF ) F iM ≤ ˜iM ≤ iM . 2 2

(3.1)

Given a closed geodesic c(t), 0 ≤ t ≤ 1, let c(t0 ) denote the cut point of c(0) along c. Then we have L(c) = d(c(0), c(t0 )) + L(c|[t0 ,1] ) ≥ d(c(0), c(t0 )) + d(c(t0 ), c(0)) ≥ 2 ˜iM .

(3.2)

Hence, (3.1) together with (3.2) implies that in order to prove the theorem, we just need to show that there (1+λ−1 )π √F exists a simple closed geodesic c with L(c) = 2 ˜iM in the case of ˜iM < . Suppose ˜iM <

2 k

(1+λ−1 )π √F . 2 k

We now construct a simple geodesic loop c (based at c(0)) with L(c) = 2 ˜iM . ˜ q) = ˜ip . Since M is compact, there is a point p ∈ M with ˜ip = ˜iM . Let q be the point in Cutp with d(p, Since d(p, q) ≤

2 π 2 ˜ ˜ √ , −1 d(p, q) = −1 ip < (1 + λF ) (1 + λF ) k

q is not the conjugate point of p. Thus, there exist two distinct normal minimal geodesics c1 (t) and c2 (t), t ∈ [0, d(p, q)] from p to q. Let c3 (t), t ∈ [0, d(q, p)] be a normal minimal geodesic from q to p. The proof of [17, Lemma 9.4] implies that either c1 ∗c3 or c2 ∗c3 is smooth at q. Without loss of generality, we assume that c1 ∗ c3 is smooth at q and therefore, c1 ∗ c3 is a geodesic loop based at p though q with ˜ q) = 2 ˜iM . We now show that c(t) = c1 ∗ c3 (t), t ∈ [0, 2 ˜iM ] is a closed L(c) = d(p, q) + d(q, p) = 2d(p, geodesic. The continuity of the cut value [3, Proposition 8.4.1] implies that for a small positive number ε(< d(p, q)), there exists tε ∈ (d(p, q), 2 ˜iM ) such that qε = c(tε ) is a cut point of pε = c(ε) along c(t). Hence, ˜ ε , qε ) ≤ d(pε , qε ) + d(qε , p) + d(p, pε ) 2 ˜iM ≤ 2d(p = d(pε , q) + d(q, qε ) + d(qε , p) + d(p, pε ) = d(p, pε ) + d(pε , q) + d(q, qε ) + d(qε , p) = d(p, q) + d(q, p) = L(c) = 2 ˜iM ,

(3.3)

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which implies that (see (3.3)) d(qε , pε ) = d(qε , p) + d(p, pε ) = L(c3 |[tε −d(p,q),d(q,p)] ) + L(c1 |[0,ε] ). That is, c = c1 ∗ c3 is smooth at p.

2

In [19], Shen introduces T-curvature, which is an important non-Riemannian quantity. However, the definition of the bound on T-curvature seems a little complicated. For convenience, we give a new definition of the bound on T-curvature. Also refer to [19,27] for more details. Definition 3.2. Given y, v ∈ Tx M with y = 0, define the T-curvature T as Ty (v) := gy (∇Vv V, y) − gy (∇Yv V, y), where V (resp. Y ) is a vector field with Vx = v (resp. Yx = y). Set Tp :=

sup y,v∈Sp M

|Ty (v)|, TM := sup Tp . p∈M

Clearly, for a compact Finsler manifold, TM is finite. And TM = 0 if and only if F is Berwaldian. By the proof of [27, Theorem 1.1], we have the following result. Theorem 3.3. Let (M, F ) be a compact Finsler n-manifold with KM ≥ k, TM ≤ τ , ΛF ≤ Λ and diam(M ) ≤ D. Then for any simple closed geodesic γ, L(γ) ≥

cn−2 Λ

3n 2

sn−1 k

μ(M )



  π min D, 2√ k n−1

1 2

+Λ τ

D

,

sn−1 (t)dt k 0

where μ(M ) is either the Busemann–Hausdorff volume or the Holmes–Thompson volume of M and cn−2 := Vol(Sn−2 ). Theorem 3.1 together with Theorem 3.3 then yields the following Cheeger type estimate. Theorem 3.4. Let (M, F ) be a compact Finsler n-manifold with |KM | ≤ k, TM ≤ τ , ΛF ≤ Λ, diam(M ) ≤ D and μ(M ) ≥ V , where μ(M ) is either the Busemann–Hausdorff volume or the Holmes–Thompson volume of M . Then ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ 1 ⎨ (1 + Λ− 2 )π ⎬ μ(M ) 1

 √ , iM ≥ . 1 min

D n−1 ⎪ ⎪ 3n 1 sn−1 1 + Λ2 k −k (D) ⎪ ⎪ 2τ ⎩ ⎭ cn−2 Λ 2 + Λ s (t)dt −k n−1 0 Remark 1. By Theorem 3.1 and the standard arguments (see [1,15]), one can show the following result, which is an extension of the results in [13,17,18]. Let (M, F ) be an even-dimensional, compact Finsler manifold with 0 < KM ≤ k. (1) If M is orientable, then iM ≥

ConjM π √ . ≥ λF λF k

In particular, if F is reversible, then iM = ConjM .

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(2) If M is not orientable, then iM ≥

π √ . λF (1 + λF ) k

4. The convex radius of a Berwald manifold Recall that a subset A ⊂ M is called strongly (geodesically) convex if for any p, q ∈ A, there exists a geodesic γpq such that γpq is the unique minimizer in M from p to q, and γpq is the only geodesic contained in A from p to q. Definition 4.1. Let (M, F ) be a forward complete Finsler manifold. The convexity radius at a point x ∈ M is defined by Convx := sup{r > 0 : Bx+ (s) is strongly convex for any s < r}. And the convexity radius of (M, F ) is defined by Conv(M, F ) := inf x∈M Convx . In [19], Shen estimates convexity radii in the reversible Finslerian case. Here, we give an estimate on the convexity radius of a Berwald manifold. Theorem 4.2. Let (M, F ) be a forward complete Berwald manifold with KM ≤ k, iM ≥ ς and λF ≤ λ. Then  Conv(M, F ) ≥ min

ς π √ , 2 k λ(1 + λ)

.

π Proof. Choosing an arbitrary point x ∈ M and any r ∈ (0, min{ 2√ , ς }), we now show that Bx+ (r) is k λ(1+λ) strictly convex. For each two points p1 , p2 ∈ Bx+ (r), let γp1 p2 (t), 0 ≤ t ≤ l denote a normal minimal geodesic from p1 to p2 . Since

l = L(γp1 p2 ) ≤ d(p1 , x) + d(x, p2 ) ≤ λ · d(x, p1 ) + d(x, p2 ) < ς/λ, γp1 p2 is the unique minimal geodesic from p1 to p2 and hence, ρ(·) := d(x, ·) is smooth on γp1 p2 ([0, l]) − {x}. Fix a point p ∈ Bx+ (r) and set Cop := {q ∈ Bx+ (r) : γpq ([0, l]) ⊂ Bx+ (r)}. We first prove that Cop is an open subset of Bx+ (r). For any sequence {qn } ⊂ Bx+ (r) − Cop converging to some point q ∈ Bx+ (r), there exists tn ∈ (0, l) such that ρ(γpqn (tn )) ≥ r for each n. Since {γpqn } is uniformly bounded, by the Arzelá–Ascoli theorem [5], we can assume that {γpqn } converges to the minimal geodesic γpq and tn → t0 . Clearly, ρ(γpq (t0 )) ≥ r, which implies q ∈ Bx+ (r) − Cop . Hence, Bx+ (r) − Cop is a closed subset of Bx+ (r). Secondly, we prove that Cop is a closed subset of Bx+ (r). It suffices to show ∂Cop ⊂ Cop . Given any point q ∈ ∂Cop , the argument is divided into the following two cases: Case 1. Suppose x ∈ / γpq . Then ρ ◦ γpq (t) is smooth, and the Hessian comparison theorem [19] implies that d2 dt2 ρ ◦ γpq (t) ≥ 0, which implies that ρ ◦ γpq (t) ≤ max{ρ ◦ γpq (0), ρ ◦ γpq (1)} < r. Hence, γpq ([0, 1]) ⊂ Bp+ (r) and therefore, q ∈ Cop .

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Case 2. If there exists t0 ∈ [0, l] such that γpq (t0 ) = x, then for t ∈ [t0 , l], ρ(γpq (t)) = L(γpq |[t0 ,t] ) ≤ L(γpq |[t0 ,l] ) = ρ(q) < r. On the other hand, one can easily find a number s0 ∈ [0, t0 ) with ρ(γpq (t)) < r for all t ∈ [s0 , t0 ]. And the argument of Case 1 implies that γpq ([0, s0 ]) ⊂ Bp+ (r). Hence, γpq ([0, 1]) ⊂ Bp+ (r) and therefore, q ∈ Cop . Since x ∈ Cop , we have Cop = Bx+ (r) and hence, Bx+ (r) is strictly convex. 2 Remark 2. Denote by TsM the upper bound of T-curvature in the sense of Shen [19]. Using the argument above, one can obtain an estimate on the convexity radius of a general Finsler manifold. More precisely, let (M, F ) be a forward complete Finsler manifold with KM ≤ k, iM ≥ ς, λF ≤ λ and TsM ≤ ξ. Then  Conv(M, F ) ≥ min v,

ς λ(1 + λ)

,

where v is the first positive zero of the following equation sk (t) − ξ · sk (t) = 0. This estimate coincides with Shen’s result [19, Theorem 15.2.1] in the reversible case. Proposition 4.3. √ Let (M, F ) be a forward complete Berwald manifold with KM ≤ k and iM ≥ ς. Set l := min{π/(2 k), ς}. Given any x ∈ M and any 0 < r < l, if a geodesic γ is tangent to the forward sphere Sx+ (r) = ∂Bx+ (r) at q, then there exists a small neighborhood Uq of q such that Uq ∩γ is outside Bx+ (r)−{q}. Proof. Suppose that γ is a normal geodesic. Let ρ(·) := d(x, ·). Clearly, for any p ∈ Bx+ (r) − {x}, ρ(p) is smooth. Set γ(t0 ) := q. Since ∇ρ is the normal vector field along Sp+ (r), we have  d  ρ(γ(t)) = g∇ρ (∇ρ, γ(t ˙ 0 )) = 0. dt t=t0

(4.1)

ρ ◦ γ(t0 ) = r < l together with Hessian comparison theorem yields that there is a small number  > 0 such d2 that ρ ◦ γ(t) < l and dt 2 ρ(γ(t)) > 0 for t ∈ (t0 − , t0 + ). This together with (4.1) implies that ρ ◦ γ has a minimum at t0 and hence, the proposition follows. 2 In the rest of this section, we assume that (A, dm) is a measure space of volume 1, and (M, F ) is a forward complete Berwald n-manifold. Given p ∈ M and r > 0, any measurable map f : A → Bp+ (r) is called a mass distribution on Bp+ (r). Define a vector field V on Bp+ (r) by V (x) := −

exp−1 x f (a) dm(a).

A

Then we have the following theorem. Refer to [11] for the results of the center of mass in the Riemannian case. Theorem 4.4. Let (M, F ) be a forward complete Berwald n-manifold with |KM | ≤ k, ΛF ≤ Λ, and iM ≥ ς. There exists a constant r = r(n, k, Λ, ς) > 0 such that for each 0 < r < r, each p ∈ M and each mass distribution f : A → Bp+ (r), there exists a unique point q ∈ Bp+ (r) with V (q) = 0. q is called the center of mass C (f ) of f .

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In particular, V (q) is differentiable and the map V∗q is non-degenerate at q = C (f ), where V∗ : T Bp+ (r) → T Bp+ (r) is defined by V∗ (X) = X i

∂V k ∂ ∂ , ∀X = X i i i k ∂x ∂x ∂x

and (xi , y i ) is a local coordinate system of T Bp+ (r). Proof. Let r :=

1 2Λ

π 1 min{ 2√ , ς√ , t, 40Λ 2 }, where t is as in Lemma Appendix A.7. Given p ∈ M and f , we k 1+ Λ

consider V (x) defined on Bp+ (r). Step 1. First, we show that for each x ∈ ∂Bp+ (r), V (x) is a nonzero outward vector. For each a ∈ A, set + Xa := − exp−1 x f (a). It is easy to see that the geodesic γXa (t), t ∈ (−, 0) is contained in Bp (r) and γXa (t), t ∈ (0, ) is outside Bp+ (r). Set ρ(·) = d(p, ·). Thus, Proposition 4.3 yields  d  ρ(γXa (t)) = g∇ρ(x) (∇ρ(x), Xa ) ≥ 0. dt t=0 Hence, ⎛ g∇ρ(x) (∇ρ(x), V (x)) = g∇ρ(x) ⎝∇ρ(x),



⎞ Xa dm(a)⎠ > 0,

A

which implies that V (x) is a nonzero outward vector. Step 2. Now we show that V has only isolated singularities in Bp+ (r). Given a ∈ A and a geodesic γ(s), s ∈ [0, 1] in Bp+ (r), consider the geodesic variation σa (t, s) = expγ(s) (t − 1)Xa , t ∈ [0, 1], where Xa := − exp−1 γ(s) f (a). Clearly, σa (0, s) = f (a) and σa (s, 1) = γ(s). Note that Us;a (t) =

∂ σa (t, s) ∂s

is a Jacobi field with Us;a (0) = 0 and Us;a (1) = γ(s). ˙ Set Ts;a (t) :=

∂ σa (t, s) = (expγ(s) )∗(t−1)Xa Xa . ∂t

It is easy to see that  Us;a (1) = ∇Ts;a Us;a = ∇Us;a Ts;a = ∇Us;a Xα .

Thus, Lemma Appendix A.7 together with the equalities above implies that            γ(s) ˙ − ∇γ(s) V = ( γ(s) ˙ − ∇ X ) dm(a) Us;a α ˙   γ(s) ˙   A γ(s) ˙ 1 1 γ(s) ˙ γ(s) ˙ ≤ dm(a) = , γ(s) ˙ γ(s) ˙ 20 20 A

which implies that V has only isolated singularities. Here · γ(s) := ˙

 gγ(s) (·, ·). ˙

(4.2)

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Step 3. We now show that V (x) has exactly one singularity in Bp+ (r). Since Bp+ (r) is contractible and V is an outward vector field along the boundary, the sum of index of V in Bp+ (r) is +1, which implies that V has at least one isolated singularity in Bp+ (r). On the other hand, for each isolated singularity z in Bp+ (r), let γ(s) be a geodesic from z. Eq. (4.2) implies that  d  gγ(s) (γ(s), ˙ V (γ(s))) = gγ(0) (γ(0), ˙ ∇γ˙ V ) > 0, ˙ ˙ ds s=0 which yields a small number l > 0 such that V is outward along ∂Bz+ (l). The Poincaré–Hopf theorem then implies that the index of V at z is +1 and therefore, V has exactly one zero in Bp+ (r). Step 4. From above, one can see that V (x) is differentiable at every point x ∈ Bp+ (r), and ∇X V = 0 for any X ∈ TC (f ) M − {0}. Let (xi , y i ) be a coordinate system of T Bp+ (r) and let γ(t), t > 0 be a smooth curve from C (f ) with γ(0) ˙ = X. Thus,

 dV i ∂ i j k + Γjk (γ(0))γ˙ (0)V (C (f )) 0 = ∇X V = dt ∂xi  ∂V i  ∂ = γ˙ k (0) i = V∗C (f ) (X), ∂xk C (f ) ∂x which implies that V∗C (f ) is nonsingular. 2 5. A Cheeger finiteness theorem for Berwald manifolds Given n ∈ N, Λ ≥ 1, ς > 0 and k ≥ 0, let r := r(n, k, Λ, ς) and C := C(n, k, Λ) be defined as in Theorem 4.4 and Lemma Appendix B.3, respectively. Definition 5.1. We say a triple (R, ε1 , ε2 ) satisfies Condition (Δ) if (1) 0 < R ≤ min



 r , C , C , C , 0 1 2 40 · Λ4

R , ε2 > 0, 12Λ3 22n+6 Λ4n+6 (1 − kR2 ) √ − C (n, k, Λ, R, ε ) − (3) ε1 > 0, 3 2 Λ5 sk ( ΛR)

(2) 0 < ε1 ≤

where  5 s−k (3Λ 2 t) ≤2 , C0 (k, Λ) := sup t > 0 : 5 3Λ 2 t  

Λt n−1 s−k (s)ds 0 2 n C1 (n, k, Λ) := sup t > 0 : t , ≤ 2(4Λ ) n−1 4Λ s (s)ds k 0   3 sk (Λ 2 t) t 2 √ C2 (k, Λ) := sup t > 0 : ≥ 1 − kt , s−k (t) s−k ( Λt) ! √ √ 6Λ3 R s−k ( ΛR) s−k ( ΛR) √ √ √ C3 (n, k, Λ, R, ε2 ) := −1 + 30Λ3 C(n, k, Λ)R2 + Λε2 . sk ( ΛR) ΛR sk ( ΛR) 

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In the following, we assume that (R, ε1 , ε2 ) is given and satisfies Condition (Δ). Definition 5.2. Given N ∈ N, we say a compact Berwald n-manifold (M, F ) satisfies Condition (1-N ) if (1) ΛF ≤ Λ, |KM | ≤ k, iM ≥ ς, diam(M ) ≥ D; 3 (2) M can be covered by N convex balls of radius R/(2Λ 2 ) 3

{Bp+α (R/(2Λ 2 )) : α = 1, . . . , N } and such balls {Bp+α (R/(4Λ2 ))}N α=1 are disjoint. 3

Let (Mi , Fi ), i = 1, 2 be two Berwald n-manifolds satisfying Condition (1-N ). Let {Bp+i (R/(2Λ 2 )) : α = α 1, . . . , N } be the forward convex balls of Mi as in Definition 5.2. Let (Rn , · ) be a standard Euclidean space. For each i, denote by · i the average Riemannian norm on Mi induced by Fi , which yields a linear isometry uiα : (Rn , · ) → (Tpiα Mi , · i ) for each α ∈ {1, . . . , N } such that for F (ui (X)) √ 1 √ ≤ i α ≤ Λ, ∀ X ∈ Rn . X Λ √ Clearly, uiα : B0 (R) → Bp+i ( ΛR), where B0 (R) (resp. Bp+i (R)) denotes the ball of radius R centered at the α α origin in (Rn , · ) (resp. (Tpiα Mi , Fi )). Set √ φiα := exppα ◦ uiα : B0 (R) → Bp+i ( ΛR). α

√

Clearly, φiα (B0 (R)) ⊃ Bp+i (R/ Λ). In particular, if φiα (B0 (R)) ∩ φiβ (B0 (R)) = ∅, the triangle inequality then α yields that φiβ (B0 (R)) ⊂ φiα (B0 (3Λ2 R)), φiα (B0 (R)) ⊂ φiβ (B0 (3Λ2 R)). Hence, for any α, β with φiα (B0 (R)) ∩ φiβ (B0 (R)) = ∅, we can define a map fiβα := (φiβ )−1 ◦ φiα : B0 (R) → B0 (3Λ2 R). The following lemma follows from Lemma Appendix A.1 directly. Lemma 5.3. There exits a constant C = C (ς, k, Λ) such that for any α, β with φiα (B0 (R)) ∩ φiβ (B0 (R)) = ∅, we have fiβα C 1 ≤ C . i For any two α, β with φiα (B0 (R)) ∩ φiβ (B0 (R)) = ∅, there exists a unique minimal geodesic γαβ (t), i i i i 0 ≤ t ≤ 1 from pα to pβ . Let Pαβ denote the parallel transformation along γαβ from Tpiα Mi to Tpiβ Mi . Define a linear isomorphism giβα : Rn → Rn by i giβα := (uiβ )−1 Pαβ uiα . i Since Fi is Berwaldian, Fi (Pαβ Y ) = Fi (Y ). Thus, one has the following result.

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Lemma 5.4. For any α, β with φiα (B0 (R)) ∩ φiβ (B0 (R)) = ∅, we have giβα X ≤ Λ. X X=0

giβα 0 := sup

By the Arzelá–Ascoli theorem, one can easily show the following lemma. Lemma 5.5. Let C be as in Lemma 5.3 and let (Rn , · ) be a Euclidean space. Set H1 := {f : B0 (R) → B0 (3Λ2 R) : f is a embedding map with f C1 ≤ C }, H2 := {f : (Rn , · ) → (Rn , · ) : f is a linear map with f 0 ≤ Λ}. Then H1 and H2 are totally bounded. That is, for each ε > 0, Hi , i = 1, 2 can be covered by a finite number of balls of radius ε. Definition 5.6. Given N ∈ N, we say two compact Berwald n-manifolds (Mi , Fi ), i = 1, 2 satisfy Condition (2-N ) if (1) (Mi , Fi ), i = 1, 2 satisfy Condition (1-N ); (2) φ1α (B0 (R)) ∩ φ1β (B0 (R)) = ∅ ⇔ φ2α (B0 (R)) ∩ φ2β (B0 (R)) = ∅, for all α, β ∈ {1, . . . , N }. The following result is a generalized Peter’s lemma, which will be proved in next section. Also refer to [14] for the original Peter’s lemma. Lemma 5.7. Let (R, ε1 , ε2 ) be a triple satisfying Condition (Δ) and let (Mi , Fi ), i = 1, 2 be two closed Berwald manifolds satisfying Condition (2-N ). Suppose that for any α, β ∈ {1, . . . , N } with φiα (B0 (R)) ∩ φiβ (B0 (R)) = ∅, we have f1βα − f2βα C1 ≤ ε1 , g1βα − g2βα 0 ≤ ε2 . Then M1 and M2 are diffeomorphic. By Lemma 5.7, we now show the following theorem. Theorem 5.8. Given n ∈ N, Λ ≥ 1, k ≥ 0 and V, D > 0, there exist only finitely many diffeomorphism classes of compact Berwald n-manifolds (M, F ) satisfying ΛF ≤ Λ, |KM | ≤ k, μ(M ) ≥ V, diam(M ) ≤ D,

(5.1)

where μ(M ) is either the Busemann–Hausdorff volume or the Holmes–Thompson volume of M . Proof. Theorem 3.4 yields a positive constant ς = ς(n, Λ, k, V, D) such that if a compact Berwald n-Finsler manifold (M, F ) satisfies (5.1), then iM ≥ ς. Let (R, ε1 , ε2 ) be a triple defined as in Definition 5.1, i.e., (R, ε1 , ε2 ) satisfies Condition (Δ). Suppose the theorem is not true. Then there exists an infinite sequence {(Ms , Fs )} satisfying (5.1), but {(Ms , Fs )} are not diffeomorphic mutually. 2 s For each s, let {Bp+sα (R/(4Λ2 ))}N α=1 denote the maximal family of disjoint balls of radius R/(4Λ ) in Ms . The volume comparison theorem [28] then implies

W. Zhao, Y. Shen / J. Math. Anal. Appl. 433 (2016) 1690–1717

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Ns ≤

μ(Ms ) ≤ C0 (n, Λ, D, R, k) =: N0 . minα μ(Bp+sα (R/(4Λ2 ))) 3

s It is not hard to check that {Bp+sα (R/(2Λ 2 ))}N α=1 cover Ms . Since {(Ms , Fs )} is an infinite sequence and N0 is a finite number, there must be a subsequence {(MsL , FsL )} such that all NsL ≡ N1 ≤ N0 . That is, for each L, the number of the maximal family of disjoint balls of radius R/(4Λ2 ) in MsL is equal to N1 . Thus, all the elements in {(MsL , FsL )} satisfy Condition (1-N1 ). Since N1 is finite, there must be a subsequence {(MK , FK )} of {(MsL , FsL )} such that for any (MK1 , FK1 ), (MK2 , FK2 ) ∈ {(MK , FK )},

K1 K2 K2 1 φK α (B0 (R)) ∩ φβ (B0 (R)) = ∅ ⇔ φα (B0 (R)) ∩ φβ (B0 (R)) = ∅,

for all α, β ∈ {1, . . . , N1 }. That is, all the elements in {(MK , FK )} satisfy Condition (2-N1 ). Lemma 5.5 yields that Hi can be covered by a finite number (say Ai ) of εi -balls, i = 1, 2. Hence, for each K, K −1 fK ◦ φK α ∈ H1 is in some a ε1 -ball. Since N1 , A1 are finite and {(MK , FK )} is an infinite sequence, βα = (φβ ) there exists a subsequence {(MK  , FK  )} such that for any (MK1 , FK1 ), (MK2 , FK2 ) ∈ {(MK  , FK  )}, we have K

K

fβα1 − fβα2 C1 ≤ ε1 , K

K

for any α, β ∈ {1, . . . , N1 } with φα 1 (B0 (R)) ∩ φβ 1 (B0 (R)) = ∅. Likewise, there exists a subsequence {(MK  , FK  )} of {(MK  , FK  )} such that for any (MK1 , FK1 ), (MK2 , FK2 ) ∈ {(MK  , FK  )}, we have K 

K 

K 

K 

fβα1 − fβα2 C1 ≤ ε1 , gβα1 − gβα2 0 ≤ ε2 , K 

K 

for any α, β ∈ {1, . . . , N1 } with φα 1 (B0 (R)) ∩ φβ 1 (B0 (R)) = ∅. Lemma 5.7 then implies that {(MK  , FK  )} are diffeomorphic mutually, which contradicts the definition of {(Ms , Fs )}. 2 6. Proof of Lemma 5.7 Before proving Lemma 5.7, we recall some notations  used in Section 5: · denotes a Euclidean norm on Rn and · i := g˜i (·, ·) denotes the average Riemannian norm induced by Fi . In  particular, for each α, uiα : (R, · ) → (Tpiα Mi , · i ) is a natural isometry. Given X ∈ T M i − {0}, · X := gi X (·, ·), where gi is the fundamental tensor induced by Fi . The proof of Lemma 5.7. Step 1. For each α, define a map Fα := φ2α ◦ (φ1α )−1 : φ1α (B0 (R)) → φ2α (B0 (R)). Given p ∈ φ1α (B0 (R)) ∩ φ1β (B0 (R)), we now estimate d(Fα (p), Fβ (p)). Since (φ1α )−1 (p) ∈ B0 (R), f1βα ◦ (φ1α )−1 (p) − f2βα ◦ (φ1α )−1 (p) C1 ≤ ε1 . Note that f1βα ◦ (φ1α )−1 (p) = (φ2β )−1 Fβ (p), f2βα ◦ (φ1α )−1 (p) = (φ2β )−1 ◦ Fα (p). Hence, (6.1) implies that

(6.1)

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  −1 F exp−1 ( F (p)) − exp ( F (p)) 2 2 β α pβ pβ  √    −1 −1 −1 ≤ Λ u−1 ◦ exp ( F (p)) − u ◦ exp ( F (p))  2 2 β α β β pβ pβ √   √ = Λ (φ2β )−1 Fβ (p) − (φ2β )−1 ◦ Fα (p) ≤ Λ ε1 .

(6.2)

√ Clearly, Fβ (p) ∈ φ2β (B0 (R)), that is, d(p2β , Fβ (p)) ≤ ΛR. Since φ1α (B0 (R)) ∩ φ1β (B0 (R)) = ∅, we have φ2α (B0 (R)) ∩ φ2β (B0 (R)) = ∅ and Fα (p) ∈ φ2β (B0 (3Λ2 R)). Let γ(t), t ∈ [0, 1] be a curve from Fα (p) to Fβ (p) such that exp−1 (γ(t)) is a straight line. Since p2 β

exp−1 (γ(t)) 2 ≤ 3Λ2 R, we have γ(t) ∈ Bp+2 (3Λ5/2 R) and p2 β

β

s−k (d(p2β , γ(t))) s−k (3Λ5/2 R) ≤ . d(p2β , γ(t)) t∈[0,1] 3Λ5/2 R max

Now Lemma Appendix A.2 and (6.2) implies that 5

3 s−k (3Λ 2 R) ε1 < 3Λ 2 ε1 . d(Fα (p), Fβ (p)) ≤ 3ΛR

Step 2. Let η : R+ → [0, 1] be a smooth function with |η  | ≤ 4 and ⎧ ⎪ ⎨ η(r) =

1, 0 ≤ r ≤ 12 , (0, 1), 12 < r < 1, 0, r ≥ 1.

⎪ ⎩

Given p ∈ M1 , set  ηα (p) := η

(φ1α )−1 (p) R



ηα (p) , ψα (p) := " . α ηα (p)

Thus, ηα (p) > 0 (or ψα (p) > 0) if and only if p ∈ φ1α (B0 (R)). Given p ∈ M1 , we define a vector field on M2 by

Vp (x) :=

N #

#     ψα (p) · exp−1 ψα (p) · exp−1 x Fα (p) = x Fα (p) , α∈Np

α=1

where Np := {α : ψα (p) = 0} = {α : p ∈ φ1α (B0 (R))}. Clearly, if α, β ∈ Np , we have φ1α (B0 (R)) ∩ φ1β (B0 (R)) = ∅. By Step 1, we have 3

3

d(Fα (p), Fβ (p)) < 3Λ 2 ε1 , d(Fβ (p), Fα (p)) < 3Λ 2 ε1 . 3

3

3

Hence, one can find a forward ball of radius 3Λ 2 ε1 , say B2 (3Λ 2 ε1 ), such that Fα (p) ∈ B2 (3Λ 2 ε1 ) for all 3 α ∈ Np . Define a mass distribution fp : Np → B2 (3Λ 2 ε1 ) by fp (α) := Fα (p). The measure mp on Np is defined by mp (α) = ψα (p). Then Vp (x) = α∈Np

exp−1 x (fp (α)) dmp (α).

(6.3)

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3

It follows from Theorem 4.4 that there exists a unique xp ∈ B2 (3Λ 2 ε1 ) such that Vp (xp ) = 0. Now we define a map F : M1 → M2 by F (p) = xp . It is not hard to check that F is well-defined. Set ! N N # #   V(p, x) := ηα (p) · Vp (x) = ηα (p) · exp−1 x Fα (p) . α=1

α=1

Note that V(p, x) is C 1 and V(p, F (p)) = 0 (cf. Theorem 4.4). The implicit function theorem then yields that [dF ] = −(D2 V)−1 · D1 V,

(6.4)

where Di V denotes the differential matrix of V respect to the i-th variable, i.e.,  D1 V :=

∂V ∂p



 , D2 V :=

∂V ∂x

 .

Note that D2 V =

!   ∂Vp . ηα (p) · ∂x α=1 N #

Theorem 4.4 then implies that D2 V is non-singular at x = F (p). Hence, (6.4) is well-defined. We will show that F is an imbedding. Eq. (6.4) implies that it is equivalent to show that D1 V|(p,F (p)) is non-singular. Note that V(p, x) ∈ Tx M2 for any fixed x. Let γ(t), t ∈ (−, ) be a smooth curve with γ(0) = p and γ(0) ˙ = X. Thus,  d  V(γ(t), x) = D1 V|(p,x) (X) ∈ TV(p,x) (Tx M2 ) ∼ = T x M2 . dt t=0 In the following, we always set x = F (p). Clearly, D1 V|(p,x) is non-singular if and only if  d  0 = V(γ(t), x) dt t=0    N #   dηα (γ(t))  dFα (γ(t))  −1 = · Yα (p) + ηα (p) · expx ∗F (p)   α dt dt t=0 t=0 α=1 =

N #

[X, dηα |p  · Yα (p) + ηα (p) · X, dYα |p ] ,

(6.5)

α=1

where Yα (p) :=

exp−1 x

Fα (p) ∈ Tx M2 , X, dYα |p :=



The proof of (6.5) is divided into three steps, i.e., Step 3–Step 5. Step 3. We first estimate I :=

N #

X, dηα |p  · Yα (p).

α=1



exp−1 x ∗Fα (p)

 dFα (γ(t))  .  dt t=0

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Note that dηα |p = 0 if and only if R < (φ1α )−1 (p) < R ⇔ p ∈ φ1α (B0 (R)) − φ1α (B0 (R/2)). 2 Thus, I=

#

X, dηα |p  · Yα (p),

α∈Np

where   Np := α : p ∈ φ1α (B0 (R)) − φ1α (B0 (R/2)) ⊂ Np . Recall that {Bp+1 (R/(4Λ))}N α=1 are disjoint. Thus, we have α

Np

≤

μ(Bp+ (ΛR)) minα∈Np μ(Bp+1 (R/(4Λ)))

≤

α

ΛR n−1 s (t)dt 2n 0 Λ R −k 4Λ sn−1 (t)dt k 0

≤ 22n+1 Λ4n .

R 1 √ 1 For each α ∈ Np , set Z(t) := exp−1 p1α (γ(t)) ∈ Tpα M1 . Clearly, F1 (Z(0)) = d(pα , p) ∈ ( 2 Λ ,   ˙ X = expp1α Hence, it follows from Lemma Appendix A.1 that Z(0).

(6.6) √

ΛR) and

∗d(p1α ,p)∇d(p1α ,p)

3

d(p1α , p) Λ2 R ˙ √ F1 (X), F1 (X) ≤ F1 (Z(0)) ≤Λ 1 sk (d(pα , p)) sk ( ΛR) which implies       4 4  d   |X, dηα |p | ≤   Z(t) 1  =  R dt t=0 R ≤

 Z(t) 21   2 Z(0) 1 



d dt t=0

˙ 4 Z(0) 1 Z(0) 4Λ2 1 √ ≤ F1 (X). R Z(0) 1 sk ( ΛR)

Since α ∈ Np ⊂ Np , Step 2 yields F2 (Yα (p)) = d(x, Fα (p)) < 6Λ2 ε1 , which together with (6.6) and (6.7) furnishes that F2 (I) =

#

F2 (X, dηα |p  · Yα (p)) ≤

α∈Np

22n+6 Λ4n+5 ε1 √ F1 (X). sk ( ΛR)

Step 4. We now estimate

II :=

N # α=1

ηα (p) · X, dYα |p  =

#

ηα (p) · X, dYα |p .

α∈Np

Given α ∈ Np . Since (u2α ◦ (u1α )−1 )∗ = u2α ◦ (u1α )−1 , for each Z ∈ T M 1 , we have

(6.7)

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1 1 F2 ((u2α ◦ (u1α )−1 )∗ Z) ≥ √ (u1α )−1 Z ≥ F1 (Z). Λ Λ

(6.8)

Recall that −1 2 1 −1 Yα (p) = exp−1 ◦ exp−1 x Fα (p) = expx ◦ expp2α ◦uα ◦ (uα ) p1 (p). α

Thus, (6.8) together with Lemma Appendix A.1 yields that   2 1 −1 F2 (X, dYα |p ) = F2 exp−1 )∗ ◦ exp−1 X 1 x∗ ◦ expp2α ∗ ◦(uα ◦ (uα ) p ∗ α





≥

d(x, Fα (p)) F2 expp2α ∗ ◦(u2α ◦ (u1α )−1 )∗ ◦ exp−1 p1α ∗ X Λs−k (d(x, Fα (p)))

≥

sk (F2 (u2α ◦ (u1α )−1 ◦ exp−1 d(x, Fα (p)) d(p1α , p) p1α (p))) F1 (X) −1 4 Λ s−k (d(x, Fα (p))) F2 (u2α ◦ (u1α )−1 ◦ expp1 (p)) s−k (d(p1α , p)) α

≥

3 2

sk (Λ R) 1 1 R √ F1 (X) ≥ 5 (1 − kR2 )F1 (X). Λ5 s−k (R) s−k ( ΛR) Λ

(6.9)

√ 3 1 Since {Bp+1 (R/(2Λ 2 ))}N α=1 is a covering, there exists β such that d(pβ , p) < R/(2 Λ), which implies that α " (φ1β )−1 (p) < R/2. Hence, β ∈ Np and α∈Np ηα (p) ≥ ηβ (p) = 1. We claim that F2 (X, dYβ |p ) − sup F2 (X, dYβ |p − dYα |p ) > 0,

(6.10)

α∈Np

which will be proved in Step 5. Here, F2 (X, dYβ |p − dYα |p ) := F2 (X, dYβ |p  − X, dYα |p ) . By (6.9) and (6.10), we have ⎛ F2 (II) = F2 ⎝ ≥

#

#

≥⎝

ηα (p) · X, dYα |p ⎠

α∈Np

ηα (p) · F2 (X, dYβ |p ) −

α∈Np

⎛

⎞

#

#

ηα (p) · F2 (X, dYβ |p − dYα |p )

α∈Np

⎞$

%

ηα (p)⎠ F2 (X, dYβ |p ) − sup F2 (X, dYβ |p − dYα |p ) α∈Np

α∈Np

≥ F2 (X, dYβ |p ) − sup F2 (X, dYβ |p − dYα |p ) α∈Np

≥

(1 − kR2 ) F1 (X) − sup F2 (X, dYβ |p − dYα |p ) . Λ5 α∈Np

(6.11)

In Step 5, we will show (6.10) and estimate (6.11). Step 5. To estimate F2 (X, dYβ |p − dYα |p ) for α, β ∈ Np , we just need to estimate the following three items

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  (1) F2 X, dYα |p  − PFα (p),x ◦ Pp2α ,Fα (p) ◦ u2α ◦ (u1α )−1 ◦ Pp,p1α X ;   (2) F2 PFα (p),x ◦ Pp2α ,Fα (p) ◦ u2α ◦ (u1α )−1 ◦ Pp,p1α X − X, dYα |p  ;  (3) F2 PFα (p),x ◦ Pp2α ,Fα (p) ◦ u2α ◦ (u1α )−1 ◦ Pp,p1α X  − PFβ (p),x ◦ Pp2β ,Fβ (p) ◦ u2β ◦ (u1β )−1 ◦ Pp,p1β X .

(6.12) (6.13)

(6.14)

Here, Pp,q denotes the parallel transformation along the normal minimal geodesic from p to q. We first estimate (6.12) and (6.13). Given α ∈ Np , set s1 := d(p1α , p) and s2 := d(p2α , Fα (p)). Clearly, there exists Y ∈ Tp1α M1 such that 

 expp1α

∗ρ1α (p)·∇ρ1α (p)

Y = X,

where ρ1α (·) := d(p1α , ·). Now let X := u2α ◦ (u1α )−1 ◦ Pp,p1α (X) ∈ Tp2α M2 , Y := u2α ◦ (u1α )−1 (Y ) ∈ Tp2α M2 ,   (s1 Y ) ∈ Tp M1 , l := d(x, Fα (p)), JY (s1 ) := expp1α 

∗s1 ∇ρ1α (p)



JY (s2 ) := expp2α

∗s2 ∇ρ2α (Fα (p))

(s2 Y ) ∈ TFα (p) M2 .

Note that there exists Z ∈ Tx M2 with (expx )∗ly Z =

  1 JY (s2 ) = expp2α (Y ), s2 ∗s2 ∇ρ2α (Fα (p))

where y := ∇ρx (Fα (p)) and ρx (·) := d(x, ·). Thus, we have   −1 Z = (expx )∗ly expp2α   −1 = (expx )∗ly expp2α

∗s2 ∇ρ2α (Fα (p))

∗s2 ∇ρ2α (Fα (p))

(Y )  −1 u2α ◦ (u1α )−1 expp1α 1

∗ρα (p)·∇ρ1α (p)

X

= X, dYα |p . Since Fi is Berwaldian, 



  1 (6.12) ≤F2 PFα (p),x J (s2 ) − PFα (p),x ◦ Pp2α ,Fα (p) X s2 Y    1 JY (s2 ) + F2 Z − PFα (p),x s2     1 JY (s2 ) − Pp2α ,Fα (p) Y + F2 Pp2α ,Fα (p) Y − Pp2α ,Fα (p) X ≤F2 s2    1 JY (s2 ) + F2 Z − PFα (p),x s2   √ 1     + F2 Y − X ≤ Λ  JY (s2 ) − Pp2α ,Fα (p) Y   s2 T1    √   1 −1  + Λ Z − Px,Fα (p) JY (s2 )   s2 T2

(6.15)

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where T1 is the velocity of the normal geodesic from p2α to Fα (p), and T2 is the velocity of the normal geodesic from x to Fα (p). √ −1 −1 Since Fi is Berwaldian, Pp,p ΛR, 1 = Pp1 ,p and Px,Fα (p) = PFα (p),x . And it is easy to see that s1 ≤ α α √ s2 ≤ ΛR and l < R. Thus, Lemma Appendix A.1 together with Lemma Appendix A.5 and Corollary Appendix A.6 yields ! √   2  1 Λ R ( ΛR) s −k  J (s2 ) − Pp2 ,F (p) Y  ≤ √ √ (6.16) − 1 F1 (X), α   s2 Y α sk ( ΛR) ΛR T1 ! √   Λ2 R s−k ( ΛR) √ √ F2 Y − X ≤ (6.17) − 1 F1 (X), sk ( ΛR) ΛR √        Λ2 R s−k (R) 1 s−k ( ΛR) Z − P −1  √ −1 J (s2 )  ≤ (6.18) F1 (X). x,Fα (p)  s2 Y sk (R) R sk ( ΛR) T2 By (6.15), (6.16), (6.17) and (6.18), we have 3Λ3 R √ (6.12) ≤ sk ( ΛR)

! √ √ s−k ( ΛR) s−k ( ΛR) √ √ −1 · F1 (X). ΛR sk ( ΛR)

3Λ3 R √ (6.13) ≤ sk ( ΛR)

! √ √ s−k ( ΛR) s−k ( ΛR) √ √ −1 · F1 (X). ΛR sk ( ΛR)

(6.19)

Similarly, one can show (6.20)

We now estimate (6.14). Given α, β ∈ Np , set Xα := Pp,p1α X ∈ Tp1α M1 , Xβ := Pp,p1β X ∈ Tp1β M1 , Xα := Pp1α ,p1β Xα ∈ Tp1β M1 , Xβ := u2β ◦ (u1β )−1 (Xβ ) ∈ Tp2β M2 , Xα := u2α ◦ (u1α )−1 (Xα ) ∈ Tp2α M2 , Xα := Pp2β ,p2α ◦ u2β ◦ (u1β )−1 (Xα ) ∈ Tp2α M2 , Xβ := Pp2β ,p2α ◦ u2β ◦ (u1β )−1 (Xβ ) ∈ Tp2α M2 . Thus, we have   (6.14) ≤F2 PFα (p),x ◦ Pp2α ,Fα (p) ◦ Xα − PFα (p),x ◦ Pp2α ,Fα (p) ◦ Xα   + F2 PFα (p),x ◦ Pp2α ,Fα (p) ◦ Xα − PFβ (p),x ◦ Pp2β ,Fβ (p) ◦ Xβ     ≤F2 Xα − Xα + F2 PFα (p),x ◦ Pp2α ,Fα (p) Xα − PFα (p),x ◦ Pp2α ,Fα (p) Xβ   + F2 PFα (p),x ◦ Pp2α ,Fα (p) Xβ − PFβ (p),x ◦ Pp2β ,Fβ (p) Xβ     =F2 Xα − Xα + F2 Xα − Xβ   + F2 PFα (p),x ◦ Pp2α ,Fα (p) Xβ − PFβ (p),x ◦ Pp2β ,Fβ (p) Xβ Firstly, we have

  2 1 −1 F (Xα − Xα ) = F Pp−1 (Xα ) − u2β ◦ (u1β )−1 (Xα ) 2 ,p2 ◦ uα ◦ (uα ) β α  √    ≤ Λ (u2β )−1 ◦ Pp2α ,p2β ◦ u2α ◦ (u1α )−1 (Xα ) − (u1β )−1 (Xα ) √   = Λ g2βα ((u1α )−1 (Xα )) − g1βα ((u1α )−1 (Xα )) ≤ Λ · ε2 · F1 (X).

(6.21)

(6.22)

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Secondly, Lemma Appendix B.3 yields F2 (Xα − Xβ ) ≤ Λ · F1 (Xα − Xβ ) ≤ C(n, k, Λ) · Λ3 · F1 (X) · R2 ,

(6.23)

where C(n, k, Λ) is the constant as in Lemma Appendix B.3. Since α, β ∈ Np , φ2α (B0 (R)) ∩ φ2β (B0 (R)) = ∅ and hence, d(p2α , p2β ) < 2ΛR. By Lemma Appendix B.3 again, we have   F2 PFα (p),x ◦ Pp2α ,Fα (p) Xβ − PFβ (p),x ◦ Pp2β ,Fβ (p) Xβ   ≤F2 PFα (p),x ◦ Pp2α ,Fα (p) ◦ Pp2β ,p2α Xβ − Pp2β ,x Xβ   + F2 Pp2β ,x Xβ − PFβ (p),x ◦ Pp2β ,Fβ (p) Xβ   5 ≤F2 PFα (p),x ◦ Pp2α ,Fα (p) ◦ Pp2β ,p2α Xβ − Pp2β ,x Xβ + 4C(n, k, Λ) · Λ 2 · F1 (X) · R2   ≤F2 PFα (p),x ◦ Pp2α ,Fα (p) ◦ Pp2β ,p2α Xβ − PFα (p),x ◦ Pp2β ,Fα (p) Xβ   5 + F2 PFα (p),x ◦ Pp2β ,Fα (p) Xβ − Pp2β ,x Xβ + 4C(n, k, Λ) · Λ 2 · F1 (X) · R2 ≤29Λ3 · C(n, k, Λ) · R2 · F1 (X).

(6.24)

Now by (6.21), (6.22), (6.23) and (6.24), we obtain & ' (6.14) ≤ 30Λ3 C(n, k, Λ)R2 + Λε2 F1 (X).

(6.25)

The triangle inequality then yields F2 (X, dYβ |p − dYα |p ) ≤ (6.12)β + (6.14)βα + (6.13)α ≤(6.19)β + (6.25)βα + (6.20)α = C3 (n, k, Λ, R, ε2 )F1 (X), which together with (6.9) and (6.11) yields (6.10) and

 (1 − kR2 ) F2 (II) ≥ − C3 (n, k, Λ, R, ε2 ) · F1 (X). Λ5

(6.26)

Step 3 furnishes that 1 √ 22n+6 Λ4n+5+ 2 √ F2 (− I) ≤ Λ · F2 (I) ≤ ε1 F1 (X). sk ( ΛR)

Thus, (6.5) together with (6.26) and (6.27) yields   d  F2 V(γ(t), x) = F2 (I + II) ≥ F2 (II) − F2 (−I) dt t=0 $ % (1 − kR2 ) 22n+6 Λ4n+6 √ ≥ − C3 (n, k, Λ, R, ε2 ) − ε1 · F1 (X) > 0, Λ5 sk ( ΛR) 

which implies that F∗ is nonsingular (see Step 2).

(6.27)

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Step 6. Since F is a local diffeomorphism, F : M1 → M2 is a covering projection (cf. [3, Theorem 9.2.1]). Let G : M2 → M1 be the map constructed as F . Given any point p ∈ M1 , there exists a point p1α ∈ M1 3 such that d(p1α , p) < R/(2Λ 2 ), which implies √ d(Fα (p1α ), Fα (p)) = F2 (u2α ◦ (φ1α )−1 (p)) ≤ Λd(p1α , p) < R/(2 Λ).

(6.28)

√ Since α ∈ Np , d(Fα (p), F (p)) < R/(2 Λ), which together with (6.28) yields √ d(p2α , F (p)) = d(Fα (p1α ), F (p)) < R/ Λ, that is, F (p) ∈ φ2α (B0 (R)). Set Gα := φ1α ◦ (φ2α )−1 : φ2α (B0 (R)) → φ1α (B0 (R)). The same argument as before yields that √ d(p1α , G ◦ F (p)) ≤ d(Gα (p2α ), Gα (F (p))) + d(Gα (F (p)), G (F (p))) < 2 ΛR, √ √ and therefore, G ◦ F (p) ∈ Bp+ (3 ΛR). Likewise, one can show F ◦ G (q) ∈ Bq+ (3 ΛR). That is, both G ◦ F and F ◦ G map every point to a convex neighborhood of itself and hence, they are homotopic to the identity. Now we conclude that F and G are diffeomorphisms. 2 Acknowledgments This work was supported by National Natural Science Foundation of China (No. 11501202), Tian Yuan Foundation (No. 11426108) and the Fundamental Research Funds for the Central Universities. Appendix A. Some estimates for Jacobi fields In this section, we always assume that (M, F ) is a compact Finsler n-manifold with ΛF ≤ Λ and |KM | ≤ k. Given y ∈ SM , we use γy (t) to denote the normal geodesic with γ˙y (0) = y. The following lemma follows from the Finslerian version of the Rauch theorem [3, Theorem 9.6.1] directly. Lemma Appendix A.1. For any y ∈ Sp M and X ∈ Tp M − {0}, we have 

(expp )∗ty X T π s−k (t) sk (t) ≤ , t ∈ 0, √ , ≤ t X T t 2 k where · T := gT (·, ·) and T := γ˙ y (t). Lemma Appendix A.2. Given three points p, q, x ∈ M . Let γ(s), s ∈ [0, 1] be a smooth curve from p to q π −1 such that d(x, γ(s)) < min{iM , 2√ } for all s. Set P := exp−1 x (p) and Q := expx (q). k (1) Suppose that γ(s) is a minimal geodesic from p to q. Then 1 sk (d(x, γ(s)) min F (Q − P ) ≤ d(p, q). Λ s∈[0,1] d(x, γ(s)) (2) Suppose that exp−1 x (γ(s)) is a straight line from P to Q. Then d(p, q) ≤ Λ max

s∈[0,1]

s−k (d(x, γ(s)) F (Q − P ). d(x, γ(s))

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Proof. For each s ∈ [0, 1], there exists Vs ∈ Tx M such that expx Vs = γ(s). We define a geodesic variation σ(t, s) := expp (tVs ), (t, s) ∈ [0, 1] × [0, 1]. Set T := where V˙ s :=

dVs ds .

∂σ ∂σ = (expx )∗tVs Vs , U := = (expx )∗tVs (tV˙ s ), ∂t ∂s

It follows from Lemma Appendix A.1 that

1 sk (d(x, γ(s)) min Λ s∈[0,1] d(x, γ(s))

1

1 F (V˙ s )ds ≤

0

s−k (d(x, γ(s)) F (U (1, s))ds ≤ Λ max d(x, γ(s)) s∈[0,1]

0

1 F (V˙ s )ds. 0

(1) Suppose that γ(s) is a minimal geodesic from p to q. Note that U (1, s) = γ(s). ˙ Hence, 1

1 F (V˙ s )ds,

F (Q − P ) ≤ 0

F (U (1, s))ds = d(p, q). 0

(2) Suppose that exp−1 x (γ(s)) is a straight line from P to Q. Thus, 1

1 F (V˙ s )ds,

F (Q − P ) = 0

F (U (1, s))ds ≥ d(p, q).

2

0

Recall the definition of curvature operator R of a Finsler manifold (cf. [28]): Given p ∈ M and y ∈ Sp M . Let Pt;y denote the parallel transformation along the geodesic γy (t) from Tp M to Tγy (t) M . The curvature operator R is defined by −1 R(t; y) := Pt;y ◦ RT ◦ Pt;y : y ⊥ → y ⊥ ,

where RT := RT (·, T )T and y ⊥ := {W ∈ Tp M : gy (y, W ) = 0}. The following result follows from |KM | ≤ k directly. Lemma Appendix A.3. Set R(t; y) := where · y :=



sup X∈y ⊥ −{0}

R(t; y)X y , X y

gy (·, ·). Thus, R(t; y) ≤ k.

Using Lemma Appendix A.3 and the same argument as in [6, Theorem IX. 4.1, Corollary IX. 4.3], one can show that Lemma Appendix A.4. Consider the vector equation of η(t) ∈ y ⊥ : η  + R(t, y)η = 0. If η(0) = 0, then

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η(s) − sη  (0) y ≤ η  (0) y · (s−k (s) − s) for all s > 0, where · y :=

 gy (·, ·).

Let A(t, y) be the solution of the matrix (or linear transformation) ordinary differential equation on y ⊥ : ⎧ ⎪ ⎨ ⎪ ⎩

A + R(t; y)A = 0, A(0; y) = 0, A (0; y) = I.

Then Pt;y A(t, y)X = (expp )∗ty tX, for any X ∈ y ⊥ . Now we have the following Lemma Appendix A.5. Given y ∈ Sp M and X ∈ Tp M , we have   (expp )∗ty X − Pt;y X  ≤ T where T := γ˙ y (t) and · T :=





 s−k (t) − 1 X T , t

gT (·, ·).

Proof. If X = ky, then (expp )∗ty X = Pt;y X. Hence, it suffices to show the lemma in the case of X ∈ y ⊥ . Set η := A(t; y)X. Clearly, η(0) = 0 and η  (0) = X. By Lemma Appendix A.4, we have A(t; y)X − tX T ≤ (s−k (t) − t) X T . It should be noted that W T = Pt;y W T for any W ∈ Tp M . Hence,       Pt;y A(t; y)X s−k (t)   − P − 1 X T . X ≤ t;y   t t T Corollary Appendix A.6. Given y ∈ Sp M and Y ∈ Tγ˙ y (t) M , where 0 ≤ t <   −1  (expp )−1 ∗ty Y − Pt;y Y T ≤ where T := γ˙ y (t) and · T := Proof. Since 0 ≤ t <

π √ , 2 k



t sk (t)



2 π √ . 2 k

Then

 s−k (t) − 1 Y T , t

gT (·, ·).

there exists a unique X ∈ Tp M such that Y = (expp )∗ty X.

Then Lemma Appendix A.1 together with Lemma Appendix A.5 yields that sk (t) −1 −1 −1 X − Pt;y Y T ≤ (expp )∗ty (X − Pt;y Y ) T = Y − (expp )∗ty Pt;y Y T t −1 −1 = Pt;y Pt;y Y − (expp )∗ty Pt;y Y T     s−k (t) s−k (t) −1 − 1 Pt;y − 1 Y T . ≤ Y T = t t

2

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Lemma Appendix A.7. Let γ(t), t ≥ 0 be a normal geodesic. Then there exists a positive constant t = t(n, k, Λ) such that for any Jacobi field J(t) along γ with J(0) = 0, we have J(t) − tJ  (t) T ≤ where T := γ(t) ˙ and · T :=



1 J(t) T , t ∈ [0, t], 20Λ

gT (·, ·).

Proof. Clearly, we have d gT (J(t) − tJ  (t), J(t) − tJ  (t)) ≤ 2 tJ  (t) T · J(t) − tJ  (t) T , dt which implies that d J(t) − tJ  (t) T ≤ tJ  (t) T = tRT (J, T )T T . dt Lemma Appendix A.3 implies that RT (J, T )T T = RT ◦ Pt;y AJ(0) T = R(t; y)AJ(0) T ≤ k · J(t) T . From above, we obtain d J(t) − tJ  (t) T ≤ kt J(t) T . dt

(A.1)

The Rauch comparison theorem yields J(t) T ≤ J  (0) T · s−k (t), π for t ∈ [0, 2√ ]. Eq. (A.1) then furnishes k

d J(t) − tJ  (t) T ≤ k J  (0) T · ts−k (t), dt which implies that (√ √ √ ) 1 J(t) − tJ  (t) T ≤ √ · J  (0) T · k t cosh k t − sinh k t k √ √ √ k t cosh k t − sinh k t √ · J(t) T . ≤ k · sk (t) Since √ lim

t→0+

√ √ k t cosh k t − sinh k t √ = 0, k · sk (t)

π there exists some t = t(n, k, Λ) ∈ (0, 2√ ) such that for t ∈ [0, t], k

√ 0<

√ √ 1 k t cosh k t − sinh k t √ < . 20Λ k · sk (t)

2

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Appendix B. Some estimates on Berwald manifolds In this section, we always assume that (M, F ) is a Berwald manifold with |KM | ≤ k and ΛF ≤ Λ. Lemma Appendix B.1. Given X, Y , W and T ∈ Sp M , we have |RT (X, Y, T, W )| ≤

√ 2 3 Λ 2 k(1 + Λ)2 . 3

Proof. Lemma Appendix A.3 yields that 3

|RU (X, Y, T, X)| = |gU (R(T, X)X, Y )| ≤ R(T, X)X U · Y U ≤ Λ 2 · k, where · U :=

(B.1)

 gU (·, ·). A direct calculation shows that 6RT (X, Y, T, W ) = − RT (W + X, Y, W + X, T ) + RT (W − X, Y, W − X, T ) − RT (T − X, Y, T − X, W ) + RT (T + X, Y, T + X, W ).

Then (B.1) furnishes that      W +X W +X  , Y, , T  F 2 (W + X) |RT (W + X, Y, W + X, T )| = RT F (W + X) F (W + X) 3

3

≤ Λ 2 k[F (W ) + F (X)]2 = 4Λ 2 k.      W −X W −X |RT (W − X, Y, W − X, T )| = RT , Y, , T  F 2 (W − X) F (W − X) F (W − X) √ 3 3 ≤ Λ 2 k[F (W ) + F (−X)]2 = Λ 2 k(1 + Λ)2 . Hence, we obtain |RT (X, Y, T, W )| ≤

√ 2 3 Λ 2 k(1 + Λ)2 . 3

2

Lemma Appendix B.2. Let Y (t) be a smooth vector field along a constant speed geodesic γ(t). Then d Y (t) ≤ ∇T Y , dt where ∇ is the Chern connection, T := γ(t) ˙ and · is the norm induced by the average Riemannian metric g˜. Proof. Denote by Pt the parallel transportation along γ from Tγ(0) M to Tγ(t) M . Choose a basis {ei } for Tγ(0) M . Then Ei (t) := Pt ei , 1 ≤ i ≤ n, is a basis of Tγ(t) M . For any w ∈ Tγ(0) M − {0}, we have 2 d g(γ(t),Pt w) (Ei (t), Ej (t)) = A(γ(t),Pt w) (Ei (t), Ej (t), ∇γ˙ Pt w) = 0. dt F (Pt w) Since (M, F ) is a Berwald manifold, Pt (Bγ(0) M ) = Bγ(t) M, Vol(x) = const,

(B.2)

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where Bx M := {y ∈ Tx M : F (x, y) < 1} and Vol(x) is the Riemannian volume of Sx M (see [19, Lemma 5.3.2] and [2]). Denote by (y i ) (resp. (z i )) the corresponding coordinate system in Tγ(0) M (resp. Tγ(t) M ) with respect to {ei } (resp. {Ei }). Thus, z i ◦ Pt = y i . Now (B.2) together with Stokes’ formula yields g˜γ(t) (Ei (t), Ej (t)) =

=

g(γ(t),v) (Ei (t), Ej (t))dz 1 ∧ · · · ∧ dz n v∈Bγ(t) M



n Vol(γ(t))

n Vol(γ(t))

g(γ(t),Pt w) (Pt ei , Pt ej )Pt∗ dz 1 ∧ · · · ∧ Pt∗ dz n

w∈Bγ(0) M

=



n Vol(γ(0))

g(γ(0),w) (ei , ej )dy 1 ∧ · · · ∧ dy n = g˜γ(0) (ei , ej ), w∈Bγ(0) M

which implies that 2 Y

d d Y = g˜γ(t) (Y (t), Y (t)) = 2˜ gγ(t) (∇T Y, Y ) ≤ 2 ∇T Y · Y . dt dt

2

Remark 3. For the Busemann–Hausdorff measure, the S-curvature of a Berwald manifold always vanishes (see [19]). The same argument as above implies that for the Holmes–Thompson measure, the S-curvature of a Berwald manifold also vanishes. Lemma Appendix B.3. Given three points p1 , p2 and p3 in M , let σij (t), 0 ≤ t ≤ 1 denote the minimizing constant speed geodesic from pi to pj . We construct a geodesic variation σ(s, t) : [0, 1] × [0, 1] → M : (1) σ(s, 0) = p1 and σ(s, 1) = p3 ; (2) Let p4 be the mid point in σ13 , that is, d(p1 , p4 ) = d(p4 , p3 ). Let σ24 (s), s ∈ [0, 1] be the minimal geodesic from p2 to p4 . For each s ∈ [0, 1], let σs (t), t ∈ [0, 12 ] be a constant geodesic from p1 to σ24 (s), and σs (t), t ∈ [ 12 , 1] be a constant geodesic from σ24 (s) to p3 . Hence, σs (t), t ∈ [0, 1] is a piecewise geodesic from p1 to p3 . Suppose that p1 p2 p3 ⊂ Bp+1 (R), where R < min{iM , 8√πkΛ }. Given a vector in X ∈ Tp1 M , set X13 := Pσ13 X and X123 := Pσ23 Pσ12 X, where Pσij is the parallel translation along σij . Then there exits a positive number C(n, k, Λ) such that F (X123 − X13 ) ≤ C(n, k, Λ) · F (X) · R2 . Proof. ∂ ∂ Step 1. Set T :=√ σ∗ ∂t , U := σ∗ ∂s . It should be noted that U is a Jacobi field. Since p1 p2 p3 ⊂ Bp+1 (R), we π have F (T ) ≤ 4R Λ < 2√k . Clearly,

 U

1 s, 2



    d 1 d d = σ s, = σ24 (s), d(p2 , p4 ) = F σ24 (s) . ds 2 ds ds

Hence,    √ 1 < 2R Λ. F U s, 2

(B.3)

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Note that for each fixed s ∈ [0, 1], there exists Ys ∈ Tp1 M such that 

  1 . U (s, t) = expp1 ∗2tT (s,0) tYs , t ∈ 0, 2 It follows from Lemma Appendix A.1 that F (tYs ) ≤

tΛF (T ) F (U (s, t)), sk (tF (T ))

which together with (B.3) then yields that F (Ys ) ≤ 2Λ

1 2 F (T ) F sk ( 12 F (T ))

   3 2πΛ 2 1 < √ R. U s, 2 k

Using Lemma Appendix A.1 again, we obtain that for s ∈ [0, 1] and t ∈ [0, 12 ], s−k (tF (T )) F (tYs ) ≤ 2s−k F (U (s, t)) ≤ Λ tF (T )



π √



2 k

5

Λ 2 R.

(B.4)

By considering the revised metric F˜ (y) := F (−y), the same argument then yields that (B.4) holds for s ∈ [0, 1] and t ∈ [ 12 , 1]. Step 2. Let Xt (s) =: Pσs (t) X denote the vector field on σ([0, 1] ×[0, 1]) induced by the parallel transformation along σs (t). Thus, for any fixed t ∈ [0, 1], we have

∇s Xt := ∇U Xt =

 ∂ dXti + Xtj Γijk U k . ds ∂xi

Since X0 (s) = X and X1 (s) ∈ Tp3 M , it is easy to see that lim+ ∇s Xt = 0, lim ∇s Xt = t→1−

t→0

dX1 (s). ds

(B.5)

Lemma Appendix B.2 together with (B.5) implies 1−

∇s Xt t=1− =

d ∇s Xt dt ≤ dt

0

1− ∇T ∇U Xt dt 0

1−

1− √ R(T, U )Xt dt ≤ Λ R(T, U )Xt T dt,

= 0

(B.6)

0

where · is the norm induced by the average Riemannian metric g˜. Let {ei } be a gT -orthonormal basis for a fixed tangent space. Set Z := R(T, U )Xt . Lemma Appendix B.1 together with (B.4) furnishes # # Z T ≤ gT (Z, ei ) · ei T = |RT (Xt , ei , T, U )| i

i

  #   ei T U Xt   = RT F (Xt ) , F (ei ) , F (T ) , F (U )  · F (Xt ) · F (ei ) · F (T ) · F (U ) i ≤ C1 · F (X) · R2 ,

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where C1 = C1 (n, k, Λ) is a constant. Eq. (B.6) then implies lim sup ∇s Xt U ≤ C2 · F (X) · R2 ,

(B.7)

t→1−

where C2 = C2 (n, k, Λ) is a constant. Using (B.5) and (B.7), we have 

1 F (X(1, 1) − X(0, 1)) ≤

F

 d X1 (s) ds ds

0

1 =

t→1 0

where C3 =

√

Λ · C2 .

1 √ lim− F (∇s Xt ) ds ≤ Λ lim sup ∇s Xt ds ≤ C3 · F (X) · R2 , 0

t→1−

2

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