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

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∇iλ+2∇kλ⋅gij+∇iλ⋅gjk+∇jλ⋅gik=0∇k∇j∇iλ+2∇kλ⋅gij+∇iλ⋅gjk+∇jλ⋅gik=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.