Invariant characterization of Liouville metrics and polynomial integrals

TL;DR

This paper develops an invariant-based criterion to identify Liouville metrics on surfaces and fully solves the local mobility problem for 2D metrics, determining the number of quadratic integrals of geodesic flows.

Contribution

It introduces a differential invariant criterion for Liouville metrics and completely solves Darboux's local mobility problem using invariant methods.

Findings

Established a differential invariant criterion for Liouville metrics

Solved the local mobility problem for 2D metrics in invariant terms

Applied the method to recognize other polynomial integrals of geodesic flows

Abstract

A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric possess? The method is also applied to recognition of other polynomial integrals of geodesic flows.