Rigidification of pseudo-Riemannian Manifolds by an Elliptic Equation

We study the solvability of the equation for the smooth function F, H=-k F g, on a geodesically complete pseudo-Riemannian manifold (M,g), H being the covariant Hessian of F. A similar equation was considered by Obata and Gallot in the Riemannian case for positive values of the constant k; the result was that the manifold must be the canonical sphere. In this generalized setting we obtain a range of possibilities, depending on the sign of k, the signature of the metric and the value of a certain first integral of the equation: the manifold is shown to be of constant sectional curvature or a warped product with suitable factors depending on the cases.