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

Pierre Mounoud (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: cone manifold; parallel tensor; projective Lichnerowicz conjecture

Domaines: Géométrie différentielle [math.DG]

Pierre Mounoud. Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and application.. 2009. ⟨hal-00403620⟩