Lovelock's theorem revisited

Let (X, g) be an arbitrary pseudo-Riemannian manifold. A celebrated result by D. Lovelock gives an explicit description of all second-order natural (0,2)-tensors on X that are symmetric and divergence-free. Apart from the dual metric, the Einstein tensor of g is the simplest example. In this paper, we give a short and self-contained proof of this theorem, simplifying the existing arguments by formalizing the notion of derivative of a natural tensor. (C) 2011 Elsevier B.V. All rights reserved.