Hyperbolic rank rigidity for manifolds of -pinched negative curvature

Abstract

A Riemannian manifold M has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of M lie in the interval [−1, − 1 4 ], and M is closed, we show that M is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures ≤ −1, and complements well-known results on Euclidean and spherical rank rigidity.is clear that the general case will be much more difficult, 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 flow 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 field J(t) along c: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 4).Corollary 1.4. Let M be a closed Riemannian manifold with sectional curvatures K between −1 ≤ K ≤ − 1 4 . Let µ be a probability measure of full support on the unit tangent bundle SM which is invariant and ergodic under the geodesic flow 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]. Definitions, Semicontinuity and Invariance on Stable ManifoldsLet 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 flow, 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. Recall that rk h (v) is the dimension of the subspace of v ⊥ which are the initial vectors of Jacobi fields that make curvature −1 with g t v for all t ≥ 0, and rk h (M) is the minimum of rk h (v) for v ∈ SM.Lemma 2.1. Let v be a unit vector recurrent under the geodesic flow. Suppose that rk h (v) > 0. Then there is also an unstable or stable Jacobi field making curvature -1 with g t v for all t ∈ R.Proof. Since rk h (v) > 0, there is a Jacobi field 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) J(tn) will converge to a Jacobi field Y (t) along c v (t). Note then that...

Carnot metrics, dynamics and local rigidity

“…The current paper began as a sequel to our previous work on the hyperbolic rank rigidity conjecture [CNS20], and indeed we do achieve new results towards Conjecture 1.13. Along the way, we also advance techniques surrounding the Brin-Pesin asymptotic holonomy group for frame flows.…”

“…This of course fails for hyperbolic surfaces. Nevertheless, local rigidity in the real hyperbolic case already follows from our previous paper [CNS20].…”

“…In Theorem 1.14, the conclusion still holds if the higher hyperbolic rank assumption is replaced with the assumption that some ergodic measure of full support has a Lyapunov exponent equal to 1. This follows from the following lemma, whose proof is essentially that of [CNS20,Corollary 1.4]. (See also [Con03,Theorem 1.3] for the case κ ≤ −1.)…”

“…In that paper, we also proved the special case of Conjecture 1.13 when the sectional curvatures satisfy [CNS20, Theorem 1.1]. Using ergodicity of 2-frame flows, Constantine [Con08, Corollary 1] had already characterized constant curvature manifolds by the hyperbolic rank condition for either odd-dimensional manifolds, just assuming non-positive curvature, or under strong pinching assumptions on the curvature, .…”

“…Using ergodicity of 2-frame flows, Constantine [Con08, Corollary 1] had already characterized constant curvature manifolds by the hyperbolic rank condition for either odd-dimensional manifolds, just assuming non-positive curvature, or under strong pinching assumptions on the curvature, . We refer to [CNS20] for more historical discussion, in particular of Hamenstädt’s hyperbolic rank rigidity theorem when the curvature is bounded above by –1.…”