Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics

TL;DR

This paper classifies pseudo-Riemannian metrics on closed surfaces with quadratic integrals in geodesic flows, solves the Beltrami problem, and proves the pseudo-Riemannian projective Obata conjecture in two dimensions.

Contribution

It provides a complete description of such metrics, establishes nonexistence results, and proves the projective Obata conjecture for 2D pseudo-Riemannian metrics.

Findings

Classification of metrics with quadratic integrals

Nonexistence of certain superintegrable metrics

Proof of the projective Obata conjecture in 2D

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, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.