Curvature Rigidity Through Level Sets of Lyapunov Exponents in Geodesic Flows

TL;DR

This paper proves new geometric rigidity results for manifolds based on the measure-theoretic properties of Lyapunov exponents, showing flatness or constant curvature under certain conditions, and introduces criteria to distinguish geodesic flows.

Contribution

It extends rigidity results by linking Lyapunov exponent level sets to curvature properties, providing unified proofs and new criteria for flow conjugacy without compactness assumptions.

Findings

Zero Lyapunov exponent level set full measure implies flatness.

Full measure Lyapunov level set indicates constant sectional curvature.

Curvature relationships obstruct smooth conjugacy of geodesic flows.

Abstract

In this paper, we establish new geometric rigidity results through the study of Lyapunov exponent level sets via invariant measures. First, we prove that for a manifold $M$ without focal points, if the zero Lyapunov exponent level set has full measure with respect to a fully supported invariant measure, then $M$ must be flat. This result recovers and extends a result of Freire and Ma\~n\'e (cf. [9]). Second, we prove that if the level set of the Lyapunov exponents has full measure with respect to some fully supported measure, then the sectional curvature must be constant. This advances the resolution of Conjecture 1 in [17]. Furthermore, we establish curvature relationships between manifolds with $1$-equivalent geodesic flows, yielding a new criterion to obstruct smooth conjugacy for flows on manifolds without conjugate points. Our techniques provide unified proofs for all rigidity results in [17] as corollaries, and additionally yield rigidity theorems for totally geodesic submanifolds in settings without conjugate points. Notably, several key results hold without compactness assumptions.