2-Frame flow dynamics and hyperbolic rank rigidity in nonpositive curvature

TL;DR

This paper proves hyperbolic rank rigidity for certain nonpositively curved spaces, showing they must have constant negative curvature under specific conditions, and introduces new rigidity results based on Lyapunov exponents.

Contribution

It extends hyperbolic rank rigidity results to even-dimensional manifolds with less restrictive pinching conditions and simplifies proofs by assuming strict negative curvature.

Findings

Proves constant negative curvature under given conditions in odd and even dimensions.

Provides a new rigidity result based solely on Lyapunov exponents.

Simplifies existing proofs by leveraging ergodicity of frame flows.

Abstract

This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-\Lambda^2 < K < -\lambda^2$ where $\lambda/\Lambda > >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity theorem of Hamenst\"{a}dt, subject to the pinching condition in even dimension; in all other cases it is a new result. We also present a rigidity result using only an assumption on maximal Lyapunov exponents in direct analogy with work done by Connell. The proof of the main theorem is simplified considerably by assuming strict negative curvature; in fact, in all dimensions but 7 and 8 it is then an immediate consequence of ergodicity of the $(dim(M)-1)$-frame flow. In these exceptional dimensions, recourse to the dynamics of the 2-frame flow must be made and the scheme of proof developed there can be generalized to deal with rank 1, nonpositively curved spaces.