Differential invariants for cubic integrals of geodesic flows on surfaces

TL;DR

This paper develops differential invariants to identify when a 2D surface's geodesic flow admits a third-degree integral related to a specific Birkhoff-Kolokoltsov 3-codifferential, advancing the understanding of integrable geodesic flows.

Contribution

It introduces a method to construct differential invariants that precisely detect the existence of certain third-degree integrals in geodesic flows on surfaces.

Findings

Differential invariants vanish exactly when the geodesic flow admits the specified integral.

Provides a new tool for studying integrability conditions of geodesic flows.

Enhances classification of metrics based on their integrability properties.

Abstract

We construct differential invariants that vanish if and only if the geodesic flow of a 2-dimensional metric admits an integral of 3rd degree in momenta with a given Birkhoff-Kolokoltsov 3-codifferential.