Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and application

TL;DR

This paper extends the Gallot-Tanno theorem to closed pseudo-Riemannian manifolds, showing conditions under which their cones are incomplete with constant curvature, and explores implications for metrics sharing geodesics.

Contribution

It generalizes the Gallot-Tanno theorem to pseudo-Riemannian manifolds and links cone properties to curvature and metric uniqueness.

Findings

Cones over such manifolds are incomplete with non-zero constant curvature.

Existence of metrics with different Levi-Civita connections but same unparametrized geodesics.

Extension of classical theorem to a broader geometric setting.

Abstract

In this article we extend the Gallot-Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over such a manifold admits a parallel symmetric 2-tensor then it is incomplete and has non zero constant curvature. An application of this result to the existence of metrics with distinct Levi-Civita connections but having the same unparametrized geodesics is given.