Partially hyperbolic geodesic flows

TL;DR

This paper constructs examples of partially hyperbolic geodesic flows that are not Anosov by deforming metrics on certain negatively curved spaces, and establishes conditions under which partial hyperbolicity can occur.

Contribution

It introduces new examples of partially hyperbolic geodesic flows beyond Anosov flows and proves that nonpositive curvature metrics with such flows must have rank one.

Findings

Partially hyperbolic geodesic flows can be constructed on deformed metrics of certain negatively curved spaces.

Product metrics and higher rank symmetric spaces are not partially hyperbolic.

Metrics with nonpositive curvature and partially hyperbolic geodesic flows must have rank one.

Abstract

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.