HYPERBOLIC RANK RIGIDITY FOR MANIFOLDS OF 1 4 -PINCHED NEGATIVE CURVATURE

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1 HYPERBOLIC RANK RIGIDITY FOR MANIFOLDS OF 1 -PINCHED NEGATIVE CURVATURE CHRIS CONNELL, THANG NGUYEN, RALF SPATZIER Abstract. A Riemannian manifold M has higher hyperbolic rank if every geodesic has a perpendicular Jacobi eld making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of M lie in the interval [ 1, 1 ], and M is closed, we show that M is a locally symmetric space of rank one. This partially extends work by Constantine using completely dierent methods. It is also a partial counterpart to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures 1, and complements well-known results on Euclidean and spherical rank rigidity. 1. Introduction We say that a closed Riemannian manifold M has higher hyperbolic rank if every geodesic c(t) in M has a nonzero perpendicular Jacobi eld J(t) which spans a plane of sectional curvature 1 with c (t) for all t 0 (where J(t) 0). Our notion of higher hyperbolic rank is a priori weaker than either the usual one which requires that the Jacobi elds in question make curvature 1 for t (, ) or else the version that uses parallel elds in place of Jacobi elds. In strict negative curvature these distinct formulations turn out to coincide (see Corollary 2.9). Actually, the techniques of our proofs require us to introduce the notion of higher hyperbolic rank for positive time. The main goal of this paper is the following hyperbolic rank rigidity result. Theorem 1.1. Let M be a closed Riemannian manifold of higher hyperbolic rank and sectional curvatures K between 1 K 1. Then M is a rank one locally symmetric space. In particular, if the pinching is strict then M has constant curvature 1. Constantine [Con08, Corollary 1] characterized constant curvature manifolds among those of nonpositive curvature and higher hyperbolic rank under one of two conditions: odd dimension without further curvature restrictions besides Euclidean rank 1, or even dimension provided the sectional curvatures are pinched between (.93) 2 and 1. He also showed that if one uses the stronger notion of parallel elds in place of Jacobi elds then one may relax the lower curvature bound of 1, though still requiring the same pinching in even dimensions. His method is rather dierent from ours, drawing on ergodicity results for the 2-frame ow of such manifolds. For 1 -pinched manifolds of negative curvature however, ergodicity of the frame ow has been conjectured now for over 30 years, with no avenue for an approach in sight [Bri82, Conjecture 2.6]. To overcome this diculty, we introduce entirely dierent 2010 Mathematics Subject Classication. Primary 53C2; Secondary 53C20,37D0. Supported in part by Simons Foundation grant #2102. Supported in part by NSF grants DMS and DMS

2 methods for 1 -pinched metrics. These also allow us to include rank one locally symmetric spaces in our classication above. Both Constantine's and our result are counterpoints to Hamenstädt's hyperbolic rank rigidity theorem [Ham91b]: Theorem 1.2. (Hamenstädt) Closed manifolds with sectional curvatures K 1 and higher hyperbolic rank are locally symmetric spaces of real rank 1. Compactness is truly essential in these results. Indeed, Connell found a counterexample amongst homogeneous manifolds of negative curvature whilst proving hyperbolic rank rigidity for such spaces under an additional condition [Con02]. Lin and Schmidt recently constructed non-compact manifolds of higher hyperbolic rank in [LS16] with both upper and lower curvature bounds 1 and curvatures arbitrarily pinched. In addition, their examples are not even locally homogeneous and every geodesic lies in a totally geodesic hyperbolic plane. In dimension three, Lin showed that nite volume manifolds with higher hyperbolic rank always have constant curvature, without imposing any curvature properties [LS16]. The notion of higher hyperbolic rank is analogous to those of higher Euclidean rank and spherical rank. However, there we are looking for parallel vector elds, not just Jacobi elds, along geodesics that make curvature 0 or 1 respectively. These versions of higher rank sometimes carry the designation strong. When 0, 1 or -1 are also extremal as values of sectional curvature, various rigidity theorems have been proved. In particular, we have the results of Ballmann and Burns-Spatzier in nonpositive curvature where higher Euclidean rank manifolds are shown to be locally either Riemannian products or symmetric spaces (cf. [Bal85, Bal95, BS87], Eberlein and Heber [EH90] for certain noncompact manifolds and Watkins [Wat13] for manifolds without focal points). When the sectional curvatures are less than 1, and M has higher spherical rank, Shankar, Spatzier and Wilking showed that M is locally isometric to a compact rank one symmetric space [SSW05]. Notably, there are counterexamples in the form of the Berger metrics for the analogous statements replacing parallel elds by Jacobi elds in the denition of higher spherical rank (see [SSW05]). Thus the situation for closed manifolds is completely understood for upper curvature bounds, and we have full rigidity. For lower curvature bounds, the situation is more complicated. For one, there are many closed manifolds of nonnegative curvature and higher Euclidean rank. The rst examples were given by Heintze (private communication) and were still homogeneous. More general and in particular inhomogeneous examples were constructed by Spatzier and Strake in [SS90]. For higher spherical rank and lower bound on the sectional curvature by 1, Schmidt, Shankar and Spatzier again proved local isometry to a sphere of curvature 1 if the spherical rank is at least n 2 > 0, n is odd or if n 2, 6 and M is a sphere [SSS16]. No counterexamples are known. If M in addition is Kähler of dimension at least, then M is locally isometric to complex projective space with the Fubini-Study metric. In dimension 3, Bettiol and Schmidt showed that higher rank implies local splitting of the metric, without any conditions on the curvature [BS16]. Let us outline our argument for Theorem 1.1 which occupies the remainder of this paper. All of our arguments hold for manifolds of with sectional curvature bounds 1 K < 0 for the rst four sections with the exception of Corollary. where we use strict 1 -pinching for 2

3 the stable distribution to be C 1. We show that we may assume that every geodesic c(t) has orthogonal parallel elds E with sectional curvature 1. The dimension of the latter vector space is called the strong hyperbolic rank of c. Following Constantine in [Con08, Section 5], strong rank agrees with the rank under lower sectional curvature bound 1 (cf. Proposition 2.6). Then we show in Section 2 that the regular set R of unit tangent vectors v for which rk h (v) = rk h (M) (cf. Denition 2.1) is dense and open. Additionally it has the property that if v R is recurrent then its stable and unstable manifolds also belong to R. Next in Section 3, we show that the distribution of parallel elds of curvature 1 is smooth on the regular set. Then, for bi-recurrent regular vectors, we characterize these parallel elds in Section in terms of unstable Jacobi elds of Lyapunov exponent 1. We use this to show that the slow unstable distribution extends to a smooth distribution on R. In Section 5, we prove the result under the stronger assumption of strict 1-curvature pinching as the technicalities are signicantly simpler and avoid the use of measurable normal forms from Pesin theory employed in Section 6. We are inspired here by arguments of Butler in [But15]. We construct a Kanai like connection for which the slow and fast stable and unstable distributions are parallel. The construction is much motivated by a similar one by Benoist, Foulon and Labourie in [BFL90]. We use this to prove integrability of the slow unstable distribution. This distribution is also invariant under stable holonomy by an argument of Feres and Katok [FK90], and hence denes a distribution on M. As it is integrable and π 1 (M)-invariant, we get a π 1 (M)-invariant foliation on M which is impossible thanks to an argument of Foulon [Fou9] (or the argument for Corollary. in [Ham91b].) Lastly, in Section 6 we treat the general case of non-strict 1 -curvature pinching. By a result of Connell [Con03], relying on Hamenstädt's Theorem 1.2, if M is not already a locally symmetric space, then the Lyapunov spectrum has non-uniform 2 : 1 resonance. Now we can use recent work of Melnick [Mel17] on normal forms to obtain a suitably invariant connection (cf. also Kalinin-Sadovskaya [KS17]). This allows us to prove integrability of the slow unstable distribution on almost every unstable manifold. As before we can obtain a π 1 - invariant foliation on M and nish with the result of Foulon as before. This is technically more complicated, however, because we no longer have C 1 holonomy maps. Instead we adapt an argument of Feres and Katok, to show that stable holonomy maps almost everywhere preserve the tangencies of our slow unstable foliation. To this end, we show that the holonomy maps are dierentiable with bounded derivatives, though not necessarily C 1, between good unstable manifolds. This allows us to obtain the desired holonomy invariance as in the strict 1-pinching case to nish the proof of the main theorem. In light of the above, in particular Theorem 1.1 as well as Constantine's results, we make the following Conjecture 1.3. A closed manifold with sectional curvatures 1 and higher hyperbolic rank is isometric to a locally symmetric space of real rank 1. Let us point out that the starting point of the proofs for upper and lower curvature bounds are radically dierent, although they share some common features. In the hyperbolic rank case in particular, for the upper curvature bound, we get control of the slow unstable foliation in terms of parallel elds. Hamenstädt used the latter to create Carnot metrics on the 3

4 boundary with large conformal group leading to the models of the various hyperbolic spaces. The lower curvature bound in comparison gives us control of the fast unstable distribution which is integrable and does not apparently tell us anything about the slow directions. It is clear that the general case will be much more dicult, even if we assume that the metric has negative or at least non-positive curvature. Finally let us note a consequence of Theorem 1.1 in terms of dynamics. Consider the geodesic ow g t on the unit tangent bundle of a closed manifold M. For a geodesic c M, the maximal Lyapunov exponent λ max (c), for c is the biggest exponential growth rate of the norm of a Jacobi eld J(t) along c: λ max (c) := max lim 1 log J(t). J Jacobi for c t Note that λ max (c) 1 if the sectional curvatures of M are bounded below by 1, by Rauch's comparison theorem. Given an ergodic g t -invariant measure µ on the unit tangent bundle SM, λ max (c) is constant µ-a.e.. In fact, it is just the maximal Lyapunov exponent in the sense of dynamical systems for g t and µ (cf. Section ). Corollary 1.. Let M be a closed Riemannian manifold with sectional curvatures K between 1 K 1. Let µ be a probability measure of full support on the unit tangent bundle SM which is invariant and ergodic under the geodesic ow g t. Suppose that the maximal Lyapunov exponent for g t and µ is 1. Then M is a rank one locally symmetric space. We supply a proof in Section 6. In fact, the reduction to Theorem 1.1 is identical to Constantine's in [Con08, Section 6] which in turn adapts an argument of Connell for upper curvature bounds [Con03]. Acknowledgements: We thank Ben Schmidt for alerting us to an incorrect denition in an earlier version and Karin Melnick for discussions about measurable normal forms. We are also thankful to the referee for a careful reading of the text and many helpful comments and suggestions, and in particular the issue of relating foliation and cocycle holonomies in Section 6. The third author is grateful to the Department of Mathematics at Indiana University for their hospitality while part of this work was completed. 2. Definitions, Semicontinuity and Invariance on Stable Manifolds Let M be compact manifold of negative sectional curvature, and denote its unit tangent bundle by SM. We let g t : SM SM be the geodesic ow, and denote by pt : SM M the footpoint map, i.e. v T pt(v) M. For v SM, let c v be the geodesic determined by v and let v denote the perpendicular complement of v in T pt(v) M. Denition 2.1. The hyperbolic rank of a vector v SM, denoted by rk h (v), is the dimension of a maximal vector subspace of v formed by the initial vectors of Jacobi elds that make curvature 1 with g t v for all t 0. Moreover, the hyperbolic rank of M, denoted by rk h (M), is the inmum of rk h (v) over v SM.

5 We remark that when 1 is an extremal curvature then the set in v of initial vectors of such Jacobi elds is already a subspace. Lemma 2.2. Let v be a unit vector recurrent under the geodesic ow. Suppose that rk h (v) > 0. Then there is also an unstable or stable Jacobi eld making curvature -1 with g t v for all t R. Proof. Since rk h (v) > 0, there is a Jacobi eld J(t) making curvature -1 with g t v for all t 0. First assume that J(t) is not stable. Decompose J(t) into its stable and unstable components J(t) = J s (t) + J u (t). Suppose g tn v v with t n. Then, for a suitable subsequence of t n, J(t+tn) will converge to a Jacobi eld Y (t) along c J(t n) v(t). Note then that g t+tn (v) g t v as t n. Moreover, for any t R, Y (t) is the limit of the vectors J(t + t n ) which make curvature -1 with g t+tn (v). Hence Y (t) also makes curvature -1 with g t v for any t. Also Y (t) is clearly unstable since J u (t) 0. If J(t) = J s (t) is stable, then the same procedure will produce a stable Jacobi eld Y (t) along c(t) that makes curvature 1 with g t (v) for all t R. Lemma 2.3. Suppose that rk h (M) > 0. Then along every geodesic c(t), we have an unstable Jacobi eld that makes curvature 1 with c(t) for all t R. Similarly, there is a stable Jacobi eld along c(t) that makes curvature 1 with c(t) for all t R. Proof. Since the geodesic ow for M preserves the Liouville measure µ, µ-a.e. unit tangent vector v is recurrent. By Lemma 2.2, the geodesics c v (t) have stable or unstable Jacobi elds along them that make curvature 1 with the geodesic for all t R. As µ has full support in SM, such geodesics are dense and the same is true for any geodesic by taking limits. Next we show that there are both stable and unstable Jacobi elds along any geodesic that make curvature 1 with the geodesic. Indeed, let A + SM be the set of unit tangent vectors v that have an unstable Jacobi eld along c v (t) that make curvature 1 with c v (t). Similarly, dene A SM as the set of unit tangent vectors v that have a stable Jacobi eld along c v (t) that make curvature 1 with c v (t). Note that A = A +, and that SM = A + A by what we proved above. Hence neither A + nor A can have measure 0 w.r.t. Liouville measure µ. Also, both A + and A are invariant under the geodesic ow g t. Since g t is ergodic w.r.t. µ, both A + and A must each have full measure. Now the claim is clear once again by taking limits. Denote by Λ(v, t)w the unstable Jacobi eld along g t v with initial value w v. Then we let E(v) v be the subspace of v dened as follows: w v belongs to E(v) if Λ(v, t)w makes curvature -1 with g t v for all t 0. We dene R = {v rk h (v) = rk h M}. We note that for v R and for all u SM, dim E(v) dim E(u). Lemma 2.. Suppose v R and w E(v). Then Λ(v, t)w makes curvature -1 with c v (t) for all t R and R is invariant under the backward geodesic ow. Proof. First note that for t R, the map Λ(v, t) : v (g t v), dened by w Λ(v, t)w is an isomorphism. We have by denition that Λ(v, t)e(g t v) E(v) for t > 0. Since v R, we have dim E(v) dim E(g t v). Thus Λ(v, t)e(g t v) = E(v) for t > 0. Therefore, for 5

6 w E(v), the Jacobi eld Λ(v, t)w along c v makes curvature -1 with g t v for all t R. This immediately implies the last statement. Next, let Ê(v) v be the subspace of v dened as follows: w v belongs to Ê(v) if the parallel vector eld along c v (t) determined by w makes curvature -1 with g t v for all t R (not just t 0 as in Denition 2.1). We have that Ê(v) E(v). Indeed if E(t) is a parallel vector eld along a geodesic c(t) that makes curvature -1 with c(t), then e t E(t) is an unstable Jacobi eld that again makes curvature -1 with c(t). Denition 2.5. The strong hyperbolic rank rk sh (v) of v is the dimension of Ê(v). The strong hyperbolic rank rk sh (M) of M is the minimum of the strong hyperbolic ranks rk sh (v) over all v SM. We use an argument of Constantine [Con08, Section 5] to prove: Proposition 2.6. If M is a closed manifold with lower sectional curvature bound 1, v R and w E(v), then the parallel vector eld determined by w along c v (t) makes curvature 1 for all t R. Thus for all v R, rk h (v) = rk sh (v) and Ê(v) = E(v). Proof. By Lemma 2., the unstable Jacobi eld Λ(v, t)w makes curvature 1 with c v (t) for all t R. Then Λ(v, t)w is a stable Jacobi eld along c v (t) still making curvature -1 with c v (t). Hence the discussion in [Con08, Section 5] shows that Λ(v, t)w = e t E where E is parallel along c v (t) for all t R. Clearly, E makes sectional curvature -1 with c v (t) as well. Note that E and Ê may not be continuous a priori. However, E and Ê are semicontinuous in the following sense. Lemma 2.7. If v n, v SM and v n v as n, then (1) lim n E(v n ) E(v) and rk h (v) lim sup n rk h (v n ) (2) lim n Ê(v n ) Ê(v) and rhsh (v) lim sup n rk sh (v n ). Here lim n E(v n ) simply denotes the set of all possible limit points of vectors in E(v n ), and similarly for Ê. Proof. These claims are clear. We now dene R = {v rk sh (v) = rk sh M}. Lemma 2.8. The sets R and R are both open with full measure and hence dense. Moreover, R is invariant under the geodesic ow. Proof. By Lemma 2.7, R is open. Since the geodesic ow is ergodic on SM w.r.t. Liouville measure and R is invariant under backward geodesic ow by Lemma 2., R has full measure. By Lemma 2.7, R is open and it is ow invariant by denition. Therefore the same argument applies. Corollary 2.9. If M is a closed manifold with lower sectional curvature bound 1, then rk h (M) = rk sh (M) and R R. 6

7 Proof. By Lemma 2.6, strong and weak rank agree on R which is an open dense set by Lemma 2.8. By Lemma 2.7, both weak and strong ranks can only go up outside R. The next argument is well-known and occurs in Constantine's work for example. As usual we let W u (v) denote the (strong) unstable manifold of v under the geodesic ow, i.e. the vectors w SM such that d(g t (v), g t (w)) 0 as t. We dene the (strong) stable manifold W s (v) similarly for t. Lemma If v R is backward recurrent under g t, then W u (v) R. If v R is forward recurrent under g t, then W s (v) R. Proof. Let w W u (v), then g t w approximates g t v when t large. On the other hand, since R is open, there is a neighborhood U of v in R. Since v is backward recurrent, g t v comes back to U and approximates v innitely often. Thus there is t large that g t w U R. It follows that w R as R is invariant under the geodesic ow (cf. Lemma 2.8). The argument for the forward recurrent case and stable leaf is similar. 3. Smoothness of Hyperbolic Rank Assume now that M has sectional curvature -1 as an extremal value, that is, either the sectional curvature K 1 or K 1. We want to prove smoothness of Ê on the regular set R. Our arguments below are inspired by Ballmann, Brin and Eberlein's work [Bal85] and also [Wat13]. First let us recall a lemma from [SSS16, Lemma 2.1]: Lemma 3.1. For v S p M, the Jacobi operator R v : v v is dened by R v (w) = R(v, w)v. Then w is an eigenvector of R v with eigenvalue -1 if and only if K(v, w) = 1. While we don't use it, let us mention [SSS16, Lemma 2.9] where smoothness of the eigenspace distribution of eigenvalue -1 is proved on a similarly dened regular set. Our situation is dierent as we characterize hyperbolic rank in terms of parallel transport of a vector not just the vector. To this end, we dene the following quadratic form: Let E(t) and W (t) be parallel elds along the geodesic c v (t), and set Ω T v (E(t), W (t)) = T T E(t) R gtve(t), W (t) R gtvw (t). Lemma 3.2. The parallel eld E(t) belongs to the kernel of Ω T v if and only if E(t) makes curvature -1 with c v (t) for t [ T, T ]. In consequence, if S < T, then ker Ω T v ker Ω S v. Proof. If E(t) makes curvature -1 with c v (t) for t [ T, T ], then E(t) R gtve(t) = 0 by Lemma 3.1, and hence E(t) is in the kernel of Ω T v. Conversely, if E(t) is in the kernel of Ω T v, let W (t) = E(t). Since the integrand now is 0 for all t [ T, T ], E(t) R gtve(t) = 0 and hence E(t) makes curvature -1 with c v (t), as claimed. Hence Ê(v) consists of the initial vectors of T ker Ω T v which is the intersection of the descending set of vector subspaces ker Ω T v as T increases. Hence there is a smallest number T (v) < such that Ê(v) consist of the initial vectors of ker ΩT v for all T > T (v). 7

8 Proposition 3.3. Ê is smooth on R. In particular, Ê is smooth on W s (v) (resp. W u (v) ) where v R is forward (resp. backward) recurrent. Proof. Let v R, and let v n v. We may assume that v n R since R is open. Note that T (v n ) < T (v)+1 for all large enough n. Otherwise, we could nd rk h M +1 many orthonormal parallel elds along c vn which make curvature -1 with c vn (t) for T (v) 1 < t < T (v) + 1. Taking limits, we nd rk h M + 1 many orthonormal parallel elds along c v which make curvature -1 with c vn (t) for T (v) 1 < t < T (v)+1. Therefore there exists a neighborhood U R of v such that T (u) < T (v) + 1 for all u U. Since the quadratic forms Ω T w (v)+1 are smooth on the neighborhood U of v, we see that the distribution is smooth on R. The last claim is immediate from smoothness on R and Lemma Maximal Lyapunov exponents and hyperbolic rank The geodesic ow g t : SM SM preserves the Liouville measure µ on SM, and is ergodic. Hence Lyapunov exponents are dened and constant almost everywhere w.r.t. µ. We recall that they measure the exponential growth rate of tangent vectors to SM under the derivative of g t. As is well-known, double tangent vectors to M correspond in a 1-1 way with Jacobi elds J(t), essentially since J(t) is uniquely determined by the initial condition J(0), J (0). Moreover we have (Dg t )(J(0), J (0)) = (J(t), J (t)). Thus we can work with Jacobi elds rather than double tangent vectors whenever convenient. We note that stable (resp. unstable) vectors for g t correspond to Jacobi elds which tend to 0 as t (resp. as t ). If 1 K 0, then all Lyapunov exponents of unstable Jacobi elds along the geodesic ow for any invariant measure are between 0 and 1, cf. e.g. [Bal95, ch. IV, Prop. 2.9]. Similarly, if K 1, all Lyapunov exponents have absolute value at least 1. We want to understand the extremal case better. We suppose K 1 throughout. Lemma.1. Let Ê(v) be the orthocomplement (with respect to the Riemannian metric on M) of Ê(v) in v. Then Λ(v, t) sends Ê(v) to Ê(g tv). Proof. Indeed, let E 1 (t),..., E n 1 (t) be a choice of parallel orthonormal elds along g t v and perpendicular to g t v such that {E 1 (t),..., E k (t)} forms a basis of Ê(g tv). For any w v, the formula for an unstable Jacobi eld becomes Λ(v, t)w = i f i (t)e i (t). Setting a ij = R(g t v, E i (t))g t v, E j (t), the Jacobi equation is equivalent to f j (t) + i a ij (t)f i (t) = 0. Since e t E i (t) is an unstable Jacobi eld for i k and the {E i (t)} are orthonormal, R(g t v, E i (t))g t v, E j (t) = E i (t), E j (t) = δ j i, 8

9 for all i k and any j n 1. By the symmetries of the curvature tensor, a ij = a ji and so we also have a ji (t) = a ij (t) = δ j i for either i k or j k. It follows that for all t R and all i k 0 = f i (t) + j a ij (t)f j (t) = f i (t) f i (t). Since Λ(v, t)w is unstable, lim f i(t) = 0 for all i. If w t Ê(v), then f i (0) = 0 for all i k. These two conditions together imply f i (t) = 0 for all t R and i k. Hence, Λ(v, t) leaves Ê invariant. Lemma.2. [Bal95, ch. IV, Prop. 2.9] [Con03, Lemma 2.3] Λ(v, t)w w e t for all t 0. The equality holds at a time T R if and only if the sectional curvature of the plane spanned by Λ(v, t)w and g t v is -1 for all 0 t T if and only if Λ(v, t)w = w e t W (t) where W (t) is parallel for all 0 t T. Proof. By the Rauch Comparison Theorem, Λ(v, t)w w e t and Λ (v, t)w Λ(v, t)w for all t 0 (cf. [Bal95, ch. IV, Prop. 2.9] which states a similar result for stable Jacobi elds). If equality holds at time T > 0 then Λ(v, t)w = e t w for all 0 t T. Indeed, should Λ(v, t 0 )w < e t 0 w for some 0 < t 0 < T, then we get a contradiction since Λ(v, T )w Λ(Λ(v, t 0 )w, T t 0 ) e T t 0 Λ(v, t 0 )w < e T t 0 e t 0 w = e T w. Therefore the vector eld W (t) for which Λ(v, t)w = w e t W (t) is a eld of norm 1. Hence W (t), W (t) = 0 and we have w e t (1 + W (t) 2 ) 1/2 = w (e t W (t) + e t W (t)) = Λ (v, t)w Λ(v, t)w = e t w, by the estimate above on the derivative of the unstable Jacobi eld. We see that W = 0, i.e. W is parallel as desired. That the sectional curvature between W (t) and the geodesic is -1 now follows from the Jacobi equation. By covering the unit tangent bundle with countable base of open sets that generate the topology, and applying the ergodic theorem to the Liouville measure, there is a full measure set of unit tangent vectors that comes back to all its neighborhoods with positive frequency. The argument in the next lemma is similar to that of Lemma 3. of [BBE85] and Proposition 1 of [Ham91a], but for the setting of a lower curvature bound of 1. We let E(v) denote the orthogonal complement of E(v) in v. Lemma.3. Suppose v R returns with positive frequency to all its neighborhoods under g t. Then for w E(v) = Ê(v), the unstable Jacobi eld Λ(v, t)w has Lyapunov exponent strictly smaller than 1. We have a similar statement for stable Jacobi elds and Lyapunov exponent -1. Proof. Let T > 0 be such that the dimension of parallel vector elds making curvature -1 with g t v for all 0 t T is k = rk h (M), i.e. k = dim E(v) since v R (2.9). Pick w 0 E(v) w that minimizes { : w Λ(v,T )w E(v) }. By Lemma.2, we have that Λ(v, T )w 0 e T w 0. Suppose that we have the equality Λ(v, T )w 0 = e T w 0. Then by Lemma.2, the parallel eld of w 0 along g t v makes curvature -1 with g t v for all 0 t T. Since w 0 E(v), the space of parallel elds making curvature -1 with g t v 9

10 for all 0 t T has dimension at least dim E(v) + 1 = k + 1, a contradiction. Therefore Λ(v, T )w 0 < e T w 0. Let ɛ > 0 be such that Λ(v, T )w 0 = (1 2ɛ)e T w 0. By continuity, we can choose a neighborhood U R of v such that for all u U and w E(u), we have the estimate Λ(v, T )w (1 ɛ)e T w. Since g t v visit U with a positive frequency, there are δ > 0 and T 0 > 0 such that for all S > T 0 {t [0, S] : g t v U} > δs. Now suppose that w v E(v). We note that by Lemma.1, since E(v) = Ê(v) on R, Λ(v, S)w E(g S v) for all S > 0. Then Λ(v, S)w e S (1 ɛ) [ δs T ] w. It follows that Λ(v, t)w has Lyapunov exponent strictly smaller than 1. We remark that the argument in the last proof only provides information that the unstable Jacobi elds come from parallel elds in forward time. This forced us to introduce both sets R and R and use the equality of E and Ê on R. Recall that there is a contact form θ on SM invariant under the geodesic ow. Its exterior derivative ω = dθ is a symplectic form on stable plus unstable distribution E s + E u. Also θ and hence ω are invariant under the geodesic ow, and thus every Oseledets space E λ with Lyapunov exponent λ is ω-orthogonal to all E λ unless λ = λ. Since ω is non-degenerate, ω restricted to E λ E λ is also non-degenerate for each λ. Note that Ê gives rise to unstable Jacobi elds with Lyapunov exponent 1. This immediately gives the following Corollary.. Assume that the manifold has strict 1 -pinched sectional curvature. The maximal Lyapunov spaces E 1 can be extended to be a C 1 distribution E1 u on the regular set R. The orthogonal complement (E1 u ) E s w.r.t. ω is dened and C 1 on the same set R and equals 1<λ<0 E λ almost everywhere. The analogous statements hold for E 1 (yielding E1 s) and E 1 E u = 0<λ<1 E λ a.e. as well. We will call the spaces E<1 s := (E1 u ) E s and E<1 u := (E1) s E u the extended slow stable and unstable subspaces. Similarly we call E1 s and E1 u the extended fast stable and unstable subspaces. Proof. On R, Ê is dened and smooth. On R, E is also dened and smooth. Moreover the distribution E agrees with Ê on R. We set Ω = {v R, v is forward recurrent under g t v with positive frequency}. By Lemma.3, E 1 agrees with the lift to unstable Jacobi elds of E on T SM, i.e. w E(v) is identied with the Jacobi eld Λ(v, t)w. The set Ω has full measure. Hence E 1 extends smoothly on R to a distribution E1 u. Now take the orthogonal complement (w.r.t. the form ω) to E1 u, (E1 u ) E s, on R in the stable distribution. Since the curvature is strictly 1-pinched, Es is C 1 [HP75]. As E1 u is even C, (E1 u ) E s is C 1. Since ω pairs Lyapunov spaces where dened a.e. on R, 1<λ<0 E λ (E1 u ) E s. Since ω is nondegenerate, the dimension of the latter subspace is exactly n 1 rk h (M) everywhere on R, and hence they agree. 10

11 A similar argument applies to E 1 and its perpendicular complement w.r.t. ω in the unstable subspace where now we use vectors backward recurrent with positive frequency. 5. Slow Stable Spaces and Integrability in Strict 1/-Pinching Throughout this section, we will assume that the sectional curvature is strictly 1/- pinched. In the tangent bundle T SM of the unit tangent bundle, consider the subset T R T SM, which is the union of tangent bers of SM at points in R. On T R, there is a C 1 decomposition E1 s + E<1 s + E 0 + E1 u + E<1, u where E s/u 1, E s/u <1 denote the extended stable/unstable fast and slow Lyapunov exponent distributions, respectively, dened in the last section. We will dene a special connection for which this decomposition is parallel, and use that to argue integrability of the slow unstable direction. Such connections were introduced by Kanai to study geodesic ows with smooth stable and unstable foliations in [Kan88]. Our particular construction is motivated by that of Benoist, Foulon and Labourie in [BFL90] where they classify contact Anosov ows with smooth Oseledets' decomposition. We refer to [GHL0, Denition 2.9 and Proposition 2.58] for the basic facts on ane connections we will need. We recall the formula for the contact 1-form θ: θ (x,v) (ξ) =< v, ξ 0 >, where (x, v) SM and ξ T (x,v) SM, where ξ 0 = d pt(ξ), and <, > denotes the Riemannian metric. Then the 2-form dθ has the property that dθ(e u, E u ) = dθ(e s, E s ) = 0 since dθ is invariant under the geodesic ow and shrinks E s /E u in forward/backward time respectively. We let X denote the geodesic spray, i.e., the generator of the geodesic ow which is the vector eld belonging to E 0 obtained by lifting unit tangent vectors of M to T SM horizontally. Note that θ(x ) = 1 and dθ(x, ) = 0. Proposition 5.1. There exists a unique connection on T R such that (1) θ = 0, dθ = 0, and E 0 E 0, E s/u i E s/u i for i {1, < 1}. (2) For any sections Z1, s Z<1, s Z1 u, Z<1 u of Es 1, E<1, s E1 u, E<1 u respectively, we have for i, j {1, < 1} Z s i Zj u = p E u j ([Zi s, Zj u ]), Z u i Z s j = p E s j ([Z u i, Z s j ]), X Z s/u i = [X, Z s/u i ], where the p s/u E are the projections to the E s/u j subspaces. j In addition, is invariant under the geodesic ow g t. Proof. Suppose rst that is a connection that satises the properties above. We note that dθ = 0 is equivalent with W dθ(y, Z) = dθ( W Y, Z) + dθ(y, W Z) for any vector elds W, Y, Z. And θ = 0 is equivalent with θ( Y Z) = Y θ(z) for any vector elds Y, Z. Thus θ( Y X ) = 0 and dθ( Y X, Z) = 0 for any vector elds Y, Z. It follows that X = 0. Furthermore, given a C 1 function f : R R, we have Y (fx ) = Y (f)x by the Leibniz rule. 11

12 Moreover, Z u i Z u j is uniquely determined by the condition E u i E u i and the equality Z u i dθ(z u j, Z s ) = dθ( Z u i Z u j, Z s ) + dθ(z u j, Z u i Z s ), for arbitrary section Z s of E s and i, j {1, < 1}. Similarly Z s i Zj s is uniquely determined. By linearity, Y Z is uniquely determined for all vector elds Y, Z. Conversely, we can use the above equations to dene a. It is then easy to check that satises the properties of a connection on R (cf. e.g. [GHL0, Denition 2.9]). That is invariant under the geodesic ow g t follows from the construction. Indeed, the slow and fast stable and unstable spaces are invariant under g t, (g t ) ([Y, Z]) = [(g t ) Y, (g t ) Z] and X is invariant under g t by denition. The next lemma is basically well-known (cf. e.g. [BFL90, Lemma 2.5]). Since our connection is only dened on a dense open set and not necessarily bounded we outline the proof. Since Liouville measure is ergodic for the geodesic ow g t on SM, the Lyapunov exponents γ i are dened and constant on a g t -invariant full measure set Σ in R. We can assume in addition that all v Σ are forward and backward recurrent for g t, and that the Oseledets decomposition T v R = E γi into Lyapunov subspaces E γi is dened on Σ. Thus if Z i E γi, the forward and backward Lyapunov exponents are dened and equal to γ i. Lemma 5.2. Let v Σ. If K is a geodesic ow invariant tensor and Z 1,..., Z k are vectors in T v R with Zi E γi, then K(Z 1,..., Z k ) is either zero or has Lyapunov exponent γ 1 + +γ k. Proof. There is a neighborhood U of v and C > 0 such that K(Y 1,..., Y k ) C Y 1 Y k, for any vectors Y 1,..., Y k with footpoints in the neighborhood U. Suppose that K(Z 1,..., Z k ) 0. If g t (v) U for some t > 0 then Thus, D v g t K(Z 1,..., Z k ) = K(D v g t Z 1,..., D v g t Z k ) C D v g t Z 1 D v g t Z k. 1 t log( D vg t K(Z 1,..., Z k ) ) 1 t (log(c) + log( D vg t Z 1 ) + + log( D v g t Z k )). Since v is forward recurrent, there will be a sequence of times t with g t (v) U. Thus the forward Lyapunov exponent of K(Z 1,..., Z k ) is at most γ γ k. Hence K(Z 1,..., Z k ) cannot have nonzero components in E γ if γ > γ γ k. Similarly, if g s (v) U for some s < 0 then 1 s log( D vg s K(Z 1,..., Z k ) ) 1 s (log(c) + log( D vg s Z 1 ) + + log( D v g s Z k )). Since v is backward recurrent, arguing as above, the backward Lyapunov exponent of K(Z 1,..., Z k ) is at least γ γ k. Hence K(Z 1,..., Z k ) cannot have nonzero components in E γ if γ < γ γ k. Recall that the connection is only C 1, and only dened on R. This means that the torsion tensor is only a C 0 -tensor, and the curvature tensor is not dened. However, slow and fast stable and unstable distributions are smooth on stable and unstable manifolds in R. Hence the restriction of to stable or unstable manifolds is also smooth by the construction 12

13 of. In particular, the curvature tensor of restricted to stable or unstable manifolds is well dened. Corollary 5.3. The torsion and curvature tensors of restricted to the slow Lyapunov distributions E s/u <1 and also each stable/unstable space Es/u are zero. Proof. Since is geodesic ow invariant, so are the torsion and curvature tensors. In strict 1 -pinched manifolds, the ratio of any two Lyapunov exponents lies in ( 1, 2). Thus this 2 corollary follows at points of Σ immediately from the previous lemma and the strict 1- pinching condition. Since Σ is of full measure, and therefore dense, the statements hold everywhere on R by continuity. Corollary 5.. The slow unstable Lyapunov distribution E u <1 is integrable. Proof. The slow unstable Lyapunov distribution E<1 u is invariant under the parallel transport by, by construction of. Since is at, parallel transport is independent of path. Thus we can choose canonical local parallel C 1 vector elds tangent to and spanning the distribution. On the other hand, since the restriction of torsion on unstable leaves is zero we have that the commutators of these vector elds are zero. By the Frobenius Theorem for C 1 vector elds, [Lan95, Theorem 1.1 Chapter 6], the distribution is integrable. As usual we will consider the π 1 (M)-lifts of the stable and unstable manifolds by the same notation in S M, and we will work in SM or S M as appropriate without further comment. Given v S M, the map π v : W u (v) M {c v ( )}, dened by π v (w) = c w ( ), is a C 1 dieomorphism [HP75]. For w g t W s (v) for some t R, the stable holonomy is dened as h v,w = π 1 w π v : W u (v) {πv 1 (c w ( )} W u (w) {πw 1 (c v ( ))}. Note that h v,w (x) is simply the intersection of the weak stable manifold of x with W u (w). In particular the stable holonomy maps are C 1. Indeed, the sectional curvatures of M are strictly 1 -pinched and hence the weak stable foliation is C1 [HP75]. Moreover, the stable holonomy maps h a,b are C 1 with derivative bounded uniformly in d S M(a, b) for b t g t W s (a). This follows from the fact that the unstable foliation is uniformly transversal to the stable foliation, by compactness of SM. In fact, Hasselblatt [Has9, Corollary 1.7] showed that the derivative is even Hölder continuous. We call a distribution stable holonomy invariant if it is invariant under (the derivative map of) all holonomies h v,w for all v S M and w W s (v). We will now adapt an argument by Feres and Katok [FK90, Lemma ]. Lemma 5.5. The slow unstable spaces E u <1 T R are stable holonomy invariant. Proof. First consider v Σ R and w t g t W s (v) Σ R. The distance between g t v and g t w remains bounded in forward time. Hence the derivatives of the holonomy maps h g t v,g t w are uniformly bounded for all t 0. If u E<1(v), u then u has forward Lyapunov exponent λ < 1 since v Σ. It follows that the image vector Dh v,w (u) also has forward Lyapunov exponent λ < 1 and hence belongs to E<1(w). u In particular, Dh v,w E<1(v) u E<1(w). u By continuity of Dh v,w and of the extended slow space on R the same holds for all v, w R. 13

14 We follow ideas of Butler [But15] to derive: Corollary 5.6. The slow unstable distributions are trivial. Proof. By the strict 1-pinching, the boundary M of the universal cover admits a C 1 structure for which the projection maps from points or horospheres are C 1 ([HP75]). By Lemma 5.5, the projection of the lifts of the slow unstable distribution is independent of the projection point on the horosphere. Using dierent horospheres we obtain a well-dened distribution on all of M. Note that this distribution is also invariant under π 1 (M). By Corollary 5., this distribution is integrable and yields a C 1 foliation F on the boundary M which is also π 1 (M)-invariant. Since there is a hyperbolic element of π 1 (M) which acts with North-South dynamics on M, by Foulon [Fou9, Corollaire], the foliation generated by this distribution has to be trivial. We are now ready to nish the proof of our main result. Proof of Theorem 1.1. (strict 1 -pinching case): Since the slow unstable distribution is trivial, all unstable Jacobi elds belong to E1 u. Hence all sectional curvatures are 1 on R. Since R is open dense in SM, it follows that all sectional curvatures are Non-Strictly 1 -Pinched Case In this section we extend the proof of the main theorem to the non-strictly 1 -pinched curvature case when the stable and unstable foliations are not necessarily C 1. This introduces two new diculties: the Kanai connection may not be dened and the stable holonomy maps may not be C Measurable Kanai connections. First, consider the set O SM of vectors whose smallest positive Lyapunov exponent is 1. If O has positive Liouville measure then Theorem of [Con03] implies M is locally symmetric and our theorem holds. Hence we may assume O has measure 0 and there is a ow invariant full measure set P and a ν > 0 such that for all v P the unstable Lyapunov exponents satisfy 1 + ν < 2 χ+ i (v) 1. Note that, unlike in the strict quarter-pinched case, we cannot immediately use the vanishing of the torsion of the generalized Kanai connection established in Proposition 5.1. Indeed, the construction of the generalized Kanai connection used that both stable and unstable distributions are C 1 on SM which we do not a priori know in our case. Instead, we replace the generalized Kanai connection with a similar one assembled from the ow invariant system of measurable ane connections on unstable manifolds constructed by Melnick in [Mel17]. The connections are dened on whole unstable manifolds but they are only dened for unstable manifolds W u (v) for v in a set of full measure. Moreover, the transversal dependence is only measurable. Mark that we have switched from Melnick's usage of stable manifolds to unstable manifolds. Following the notation in [Mel17], let her E be the smooth tautological bundle over SM whose ber at v is W u (v). We consider the cocycle Fv t which is g t restricted to W u (v). The ratio of maximal to minimal positive Lyapunov exponents lies in [1, 2), and hence the integer r appearing in Theorem 3.13 of [Mel17] is 1. This theorem then reads in our notation as: 1

15 Lemma 6.1. There is a full measure ow-invariant set U SM where there is a smooth ow-invariant at connection on T W u (v) for v U. Now we build a connection on vector elds tangent to the slow unstable distribution E u <1 on W u (v) for v U. We emphasize that we do not assume integrability of the slow unstable distribution. We just construct a connection on sections of the vector bundle given by the slow unstable distribution. More specically on slow unstable distribution we have the following. Lemma 6.2. On each unstable leaf W u (v) for v in a full measure ow invariant subset Q U P R, there exists a torsion free and ow invariant connection, <1 : T W u (v) Γ 1 (W u (v), E u <1) E u <1 on E<1 u. (Here Γ1 (W u (v), E<1) u represents C 1 sections.) Moreover the restriction of the connection to E<1 u is torsion free. Proof. Recall that the distribution E<1 u is smooth on W u (v) for v R. Given X T W u (v) and Y Γ 1 (W u (v), E<1) u we dene the covariant derivative <1 X Y to be the vector in E<1 u W u (v) given by projection of the Melnick connection, <1 X Y := proj E u <1 X Y. Note that this operator is R-bilinear in X and Y since projections are linear, and for f C 1 (W u (v)) since scalar functions commute with projection we have <1 fx Y = proj E u <1 f XY = f proj E u <1 X Y = f <1 X Y <1 X fy = proj E u <1 X(f)Y + f X Y = X(f)Y + f <1 X Y. Here we have used that proj E u <1 Y = Y. Hence <1 is C 1 (W u (v))-linear in X, and satises the derivation property of connections. Observe that <1 can then be extended to a map of sections <1 : Γ 1 (W u (v), E u <1) Γ 1 (W u (v), E u <1) Γ 1 (W u (v), E u <1). For v U, X, Y Γ 1 (W u (v), E u <1) the torsion tensor T (X, Y ) = <1 X Y <1 Y X [X, Y ] is indeed a tensor due to the derivation property of the connection and bracket where we take the bracket of vector elds in W u (v). Next we show that <1 is torsion free. Since [X, Y ] and <1 are invariant under Dg t, so is T (X, Y ). Also, since v P the sum of any two Lyapunov exponents lies in (1, 2]. By Fubini, and absolute continuity of the W u foliation, we may choose Q U P R to be an invariant full measure set where for each v Q a.e. w W u (v) is forward and backward recurrent. By Lusin's theorem, for all ɛ > 0, there is a compact set A of measure > 1 ɛ such that the torsion tensor T is continuous on A. In particular, there is a constant C > 0 such that for all tangent vectors X, Y at a A, T (X, Y ) C X Y. Also note that for a.e. x A, g t (x) A for innitely many t, by ergodicity of g t. Then the argument from Lemma 5.2 shows that T vanishes on A. As ɛ > 0 is arbitrary, T vanishes on a set of full measure. In addition, this set is automatically g t -invariant as desired. Corollary 6.3. The slow unstable Lyapunov distribution E<1 u W u (v) for v Q. 15 is integrable on every leaf

16 Proof. For v Q, and X, Y Γ 1 (W u (v), E<1) u the vanishing of the torsion tensor implies T (X, Y ) = 0 = <1 X Y <1 Y X [X, Y ]. However, by denition <1 X Y and <1 Y X belong to E<1, u and therefore so does [X, Y ]. In particular, E<1 u is integrable. This corollary gives us well dened slow unstable foliations on almost every W u (v). For such v, we denote its leaves by W<1(w) u for w W u (v). Note that the fast unstable distribution is also integrable with leaves we denote by W1 u (w). Next we will show that these foliations are invariant under stable holonomy. This is substantially more dicult in the non-strict 1 -pinched case since the unstable holonomy maps a priori are not known to be C Stable holonomies are C 1 a.e. We will now address the second diculty, namely that the stable holonomy maps are not globally C 1. Essentially we will approximate the stable holonomy by C 1 approximate holonomies, and show that the latter have dierentiable limits a.e.. This approach is inspired by work of Avila, Santamaria and Viana on cocycle holonomy maps in [AV10, ASV13]. These are easier to control than foliation holonomies. For us, the cocycle is the unstable derivative cocycle. We will show existence and estimates of such, and relate them to the stable foliation holonomy. The latter is similar to work of Burns and Wilkinson [BW05] and Brown [Bro16]. Our situation is a tad more technical as we only have pinching of Lyapunov exponents. To simplify notation, we use Dg t,v for the derivative of g t at v restricted to E u (v). Since M is strictly 1 + δ-pinched for any δ > 0, Corollary 1.7 of [Has9] implies the following. Lemma 6.. The foliations W s and W u are α-hölder for all α < 1. Now choose an α > 1 ν. As in Kalinin-Sadovskaya [KS13, Section 2.2] we have local linear identications I vw : E u (v) E u (w) such that I v,v = Id and Iv,w 1 = I w,v and which vary in an α-hölder way on a neighborhood of the diagonal in SM SM. We also have that Dg t is an α-hölder cocycle, since it is the restriction of the smooth Dg t to an α-hölder bundle. In other words, with respect to these identications, we have (6.1) Dg t,v Ig 1 tv,g tw Dg t,w I v,w C(T 0 )d(v, w) α for any T 0 > 0 and all t T 0. We will now construct local smooth maps approximating the stable holonomy whose derivatives have certain properties. This will allow us to connect the holonomy h v,w to linearized approximations by providing a superior choice of identications I vw : E u (v) E u (w). For v S M let W cs (v) = t R W s (g t v), the weak unstable leaf of v, and set W u (v, ɛ) = B(v, ɛ) W u (v). Lemma 6.5. There are constants ɛ 0, C > 0 such that for any v S M and w W cs (v) with d(v, w) < ɛ 0, there are C maps I v,w : W u (v, ɛ 0 ) W u (w) satisfying: (1) d(i v,w (v), w) < Cd(v, w), (2) d(di v,w (ξ), ξ) < Cd(v, w) α for all ξ S v W u, (3) DI v,w 1 < Cd(v, w) α, () if v W u (v) and w ( t R g t W s (v, ɛ 0 )) W u (w) then I v,w = I v,w on the common part of their domain. 16

17 (6.2) (5) for any T 0 > 0 and all t T 0, we have d(dg t,v (ξ), D gti v,w(v)(i gtv,g tw) 1 Dg t,iv,w(v) D v I v,w (ξ)) C(T 0 )d(v, w) α ξ, for every ξ E u (v). Proof. We consider S M with the Sasaki metric induced from M lifted to the universal cover M. By compactness of SM the operator norm of the second fundamental forms at all points of the individual smooth leaves W u (v) are bounded above by a universal constant. Hence the focal radius of each W u (v) is uniformly bounded from below by a number i 0 independent of v S M. Thus the exponential map exp for the Sasaki metric applied to the normal bundle ν to E u (v) = T v W u is injective on all of B (W u (v), t) = y W u (v)b (y, t) for all t < i 0 and v S M. (Here B (y, t) indicates the ball of radius t in the vector space ν y.) By compactness of SM and continuity of E u, there is a universal r 0 > 0 such that the leaves W u (w) will be uniformly transverse to exp y B (y, i 0 ) whenever they intersect for some y W u (v) and whenever d(v, w) < r 0. Let ɛ 0 = min{ i 0 2, r 0 2 }. The images exp y B (y, i 0 2 ) for y W u (v) foliate a normal neighborhood of W u (v). We call this the normal disk foliation. When d(v, w) < ɛ 0 dene I v,w (v ) to be the intersection of exp v B (v, ɛ 0 ) W u (w) on the maximal domain of W u (v) where the intersection exists. By construction the intersection point will be unique if it exists. In other words, I v,w is the holonomy for the normal disk foliation and thus we may extend it to every leaf exp y B (y, ɛ 0 ) of the normal disk bundle whenever this intersects W u (w). Property # now follows from the construction since the normal disk foliation is dened in a normal neighborhood of W u (v). By compactness of M, the derivatives of the normal bundle to W u (v) are bounded. Hence for any v S M and v W u (v, ɛ 0 ), t < ɛ 0, and ξ Tv W u (v) we have d(exp v (tξ), exp v (B (v, ɛ 0 )) < C d(v, v ) for some uniform constant C. Let β : [0, d W cs (v)(v, w)] W u (w) be the projection of the unit speed W cs (v)-geodesic from v to w to W u (w) under I v,w. We measure d(w, I v,w (v)) l(β) dw cs (v) (v,w) 0 C sin( (t))dt C d W cs (v)(v, w), where (t) represents the angle between the W cs (v) geodesic and the T exp y B (y, ɛ 0 ) distribution. We note that for small distances, Theorem.6 of [HeiImH] implies that the induced distance on a horosphere of M is comparable to the distance in M. It then follows from the denition of the Sasaki metric and the universal bound on the second fundamental form of the horospheres that there is a uniform constant C such that d t R g tw s (v)(v, w) C d(v, w). The rst statement now follows with C = C C. For statements #2 and #3, choose a smooth coordinate chart φ on a neighborhood U W u (v) of v W u (v) and on each y U use exp 1 y to obtain a global smooth chart ψ from a neighborhood O of v S M to R n R n 1 such that W u (v) maps to {0} R n 1. Note that for each v U, the normal disk exp v B (v, ɛ 0 ) is carried into R n {φ(v )} isometrically at v since d 0 exp v = Id and moreover the orthogonality is preserved. In particular, the map I v,w becomes the horizontal holonomy onto the image ψ(w u (w)) and D v I v,w maps to the projection of T 0 {0} R n 1 = {0} R n 1 onto T ψ(iv,w(v))ψ(w u (w)). However, since E u is α-hölder, d Gn 1 ({0} R n 1, T ψ(ηv,w(v))ψ(w u (w))) < Cd(v, w) α with respect to the distance 17