Proof of projective Obata conjecture for two-dimensional pseudo-Riemannian metrics

TL;DR

This paper proves that on closed two-dimensional pseudo-Riemannian manifolds, any projective vector field must be Killing, confirming a conjecture that extends classical results to a broader geometric setting.

Contribution

It establishes the two-dimensional pseudo-Riemannian version of the projective Obata conjecture, showing the equivalence of projective and Killing vector fields in this context.

Findings

Every projective vector field is Killing on closed 2D pseudo-Riemannian manifolds.

The result excludes the round sphere as a special case.

Confirms the conjecture for a broader class of geometries.

Abstract

I prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture stating that on a closed manifold different from the round sphere every projective (i.e., geodesic-preserving) vector field is Killing.