Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

TL;DR

This paper classifies 2D pseudo-Riemannian metrics with geodesic flows admitting quadratic integrals, providing local normal forms and demonstrating their properties and applications in integrable systems and quantum mechanics.

Contribution

It introduces local normal forms for such metrics and explores their properties, including geodesic equivalence and applications to integrable systems.

Findings

Metrics admit geodesically equivalent metrics

Construction of large families of integrable natural systems

Extension of integrability to quantum systems

Abstract

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: 1) they admit geodesically equivalent metrics; 2) one can use them to construct a big family of natural systems admitting integrals quadratic in momenta; 3) the integrability of such systems can be generalized to the quantum setting; 4) these natural systems are integrable by quadratures.