Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta

TL;DR

This paper classifies pseudo-Riemannian metrics on closed surfaces with geodesic flows that have nontrivial quadratic integrals, solves the Beltrami problem, and shows such metrics cannot be quadratically superintegrable with nonconstant curvature.

Contribution

It provides a complete description of pseudo-Riemannian metrics with quadratic integrals on closed surfaces and addresses longstanding problems in the geometry of geodesic flows.

Findings

Classification of metrics with quadratic integrals

Solution to the Beltrami problem on closed surfaces

Proof of nonexistence of certain superintegrable metrics

Abstract

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces