Curvature and partial hyperbolicity

TL;DR

This paper establishes a criterion linking the curvature of Riemannian manifolds to the partial hyperbolicity of their geodesic flows, providing examples that both satisfy and defy the criterion.

Contribution

It introduces a new quadratic form-based criterion connecting curvature and partial hyperbolicity in geodesic flows, with illustrative examples.

Findings

Some manifolds satisfy the criterion and have partially hyperbolic geodesic flows.

An example exists where the criterion is not satisfied but the flow remains partially hyperbolic.

The criterion helps identify conditions for partial hyperbolicity in Riemannian geometry.

Abstract

Using quadratic forms, we stablish a criteria to relate the curvature of a Riemannian manifold and partial hyperbolicity of its geodesic flow. We show some examples which satisfy the criteria and another which does not satisfy it but still has a partially hyperbolic geodesic flow.