On the ergodicity of geodesic flows on surfaces of nonpositive curvature

Abstract

Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\ge 2$. 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.

11 pages, 4 figures. Lemma 3.8 is added to correct a gap in the proof of Proposition 3.4. Theorem 1.5 is also added

References in corpus (1)

• Open problems and questions about geodesics

Cited by in corpus (1)

• Minimal geodesics and integrable behavior in geodesic flows