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

Vladimir S. Matveev (1) , Pierre Mounoud (2)

1 Mathematisches Institut, FSU Jena
2 IMB - Institut de Mathématiques de Bordeaux

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

Domaines: Géométrie différentielle [math.DG]

Vladimir S. Matveev, Pierre Mounoud. Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications.. Annals of Global Analysis and Geometry, 2010, 38 (3), pp.259-271. ⟨10.1007/s10455-010-9211-7⟩. ⟨hal-00420655v2⟩