On the ergodicity of geodesic flows on surfaces of nonpositive curvature

TL;DR

This paper proves the ergodicity of geodesic flows on certain nonpositively curved surfaces and establishes finiteness results for flat geodesics and strips under specific curvature conditions.

Contribution

It demonstrates ergodicity of geodesic flows on nonpositively curved surfaces with finitely many negative curvature components and proves finiteness of flat geodesics and strips.

Findings

Ergodicity of geodesic flow under specified curvature conditions

Finiteness of flat geodesics and flat strips

Non-existence of non-closed flat geodesics under the same conditions

Abstract

Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed "flat" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed "flat" geodesics.