Complete Einstein metrics are geodesically rigid

TL;DR

This paper proves that complete Einstein metrics are uniquely determined by their unparametrized geodesics, establishing their geodesic rigidity in both Riemannian and pseudo-Riemannian cases.

Contribution

It demonstrates that any other complete metric sharing the same unparametrized geodesics must have the same Levi-Civita connection, confirming geodesic rigidity for Einstein metrics.

Findings

Complete Einstein metrics are geodesically rigid.

Any other complete metric with the same unparametrized geodesics shares the same Levi-Civita connection.

The result applies to both Riemannian and pseudo-Riemannian Einstein metrics.

Abstract

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and $\bar g$ coincide.