Local normal forms for geodesically equivalent pseudo-Riemannian metrics

TL;DR

This paper provides a comprehensive local classification of pseudo-Riemannian metrics that are geodesically equivalent, extending classical results to indefinite metric signatures.

Contribution

It offers a complete local description of geodesically equivalent pseudo-Riemannian metrics, generalizing the Beltrami problem to indefinite metrics.

Findings

Complete local description of geodesically equivalent metrics

Extension of Beltrami problem to pseudo-Riemannian case

New classification results for indefinite metrics

Abstract

Two pseudo-Riemannian metrics $g $ and $\bar g$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We give a complete local description of such metrics which solves the natural generalisation of Beltrami problem for pseudo-Riemannian metrics.