Integrable geodesic flows on surfaces

TL;DR

This paper introduces a new condition to analyze integrable geodesic flows on closed surfaces, strengthening non-integrability results and characterizing phase portraits on the 2-torus.

Contribution

It proposes the condition $eth$ for studying integrability, extends Kozlov's non-integrability theorem, and describes the phase portraits of integrable flows on the 2-torus.

Findings

Strengthens Kozlov's theorem on non-integrability for higher genus surfaces.

Characterizes phase portraits of integrable flows on the 2-torus.

Shows the structure of invariant sections and separatrix chains in integrable flows.

Abstract

We propose a new condition $\aleph$ which enables to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov's theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on 2-torus. Our main result for 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside, what we call, separatrix chains. The complement to the union of the separatrix chains is $C^0$-foliated by invariant sections of the bundle.