Fractional-linear integrals of geodesic flows on surfaces and Nakai's geodesic 4-webs

TL;DR

This paper classifies fractional-linear integrals of geodesic flows on surfaces, showing their dimension depends on curvature and linking their existence to specific geometric web structures with constant cross-ratio.

Contribution

It establishes a classification of fractional-linear integrals for geodesic flows and connects their existence to geodesic 4-webs with constant cross-ratio.

Findings

Dimension of integral space is 3 or 5

Constant curvature corresponds to dimension 5

Existence of integrals linked to geodesic 4-webs with constant cross-ratio

Abstract

We prove that if the geodesic flow on a surface has an integral, fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for existence of fractional-linear integrals: such integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves.