Gallot–Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist

Abstract

We generalize for complete pseudo-Riemannian metrics a classical result of Gallot (1979) [3] and Tanno (1978) [13]: we show that if a closed complete manifold admits a nonconstant function λ satisfying ∇ k ∇ j ∇ i λ + 2 ∇ k λ ⋅ g i j + ∇ i λ ⋅ g j k + ∇ j λ ⋅ g i k = 0 , then the metric is the Riemannian metric of constant curvature +1. We use this result to give a simple proof of a recent result of Alekseevsky, Cortes, Galaev and Leistner (2009) [1]. Certain generalizations for higher Gallot equations are given.