Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications

TL;DR

This paper extends the Gallot-Tanno Theorem to closed pseudo-Riemannian manifolds, showing that certain tensor conditions imply Riemannian structure and applying this to geodesic and holonomy problems.

Contribution

It generalizes the Gallot-Tanno Theorem to pseudo-Riemannian manifolds and explores implications for geodesic equivalence and holonomy group actions.

Findings

Cone over a manifold with a parallel symmetric tensor is Riemannian.

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

Holonomy group of certain manifolds does not preserve any nondegenerate splitting.

Abstract

We extend the Gallot-Tanno Theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over a manifold admits a parallel symmetric $(0,2)-$tensor then it is Riemannian. Applications of this result to the existence of metrics with distinct Levi-Civita connections but having the same unparametrized geodesics and to the projective Obata conjecture are given. We also apply our result to show that the holonomy group of a closed $(O(p+1,q),S^{p,q})$-manifold does not preserve any nondegenerate splitting of $\R^{p+1,q}$.