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

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

minor corrections

References in corpus (3)

• Complete Einstein metrics are geodesically rigid
• Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two
• Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist

Cited by in corpus (12)

• There exist no 4-dimensional geodesically equivalent metrics with the same stress-energy tensor
• Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two
• Projectively related metrics, Weyl nullity, and metric projectively invariant equations
• Conification construction for Kaehler manifolds and its application in c-projective geometry
• Projective Structure and Holonomy in 4-dimensional Lorentz Manifolds
• Orthogonal separation of variables for spaces of constant curvature
• Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics
• (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry
• The degree of mobility of Einstein metrics
• A special class of symmetric Killing 2-tensors
• On the number of nontrivial projective transformations of closed manifolds
• Semi-Riemannian cones