Divergence theorem for symmetric (0,2)-tensor fields on a semi-Riemannian manifold with boundary

Abstract

We prove in this paper a divergence theorem for symmetric (0,2)-tensors on a semi-Riemannian manifold with boundary. We obtain a generalization of results obtained by Ünal in [9, Acta Appl. Math. 40(1995)] and E. García-Río and D. N. Kupeli in [4, Proceeding of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000). Nonlinear Anal. 47 (5) 2995-3004, 2001]. As a tool, we use an induced volume form on the degenerate boundary by introducing a star like operator. A vanishing theorem for gradient timelike Killing vector fields on Einstein semi-Riemannian manifolds is obtained.