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

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

2009-09-29

arxiv.org/abs/0909.5344v2

Annals of Global Analysis and Geometry 38, 3 (2010) 259-271

Mathematics

Differential Geometry

minor corrections

Scientific paper

10.1007/s10455-010-9211-7

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 $\R^{p+1,q}$.

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