On fractional-linear integrals of geodesics on surfaces

TL;DR

This paper establishes criteria for fractional-linear integrals in geodesic flows on surfaces, explores their moduli space structure under M"obius transformations, and relates these integrals to Killing vectors through explicit examples.

Contribution

It provides a new criterion for the existence of fractional-linear integrals and characterizes their moduli space on Riemannian surfaces.

Findings

Moduli space of such integrals is either a projective plane or finite points.

Explicit examples illustrating the integrals and their properties.

Connection between rational integrals and Killing vectors.

Abstract

In this note we give a criterion for the existence of a fractional-linear integral for a geodesic flow on a Riemannian surface and explain that modulo M\"obius transformations the moduli space of such local integrals (if nonempty) is either the two-dimensional projective plane or a finite number of points. We will also consider explicit examples and discuss a relation of such rational integrals to Killing vectors.