On second-order, divergence-free tensors Available to Purchase

The aim of this paper is to describe the vector spaces of those second-order tensors on a pseudo-Riemannian manifold (i.e., tensors whose local expressions only involve second derivatives of the metric) that are divergence-free. The main result establishes isomorphisms between these spaces and certain spaces of tensors (at a point) that are invariant under the action of an orthogonal group. This result is valid for tensors with an arbitrary number of indices and symmetries among them and, in certain cases, it allows to explicitly compute basis, using the classical theory of invariants of the orthogonal group. In the particular case of tensors with two indices, we prove the Lovelock tensors are a basis for the vector space of second-order, divergence-free 2-tensors. This statement generalizes to arbitrary dimension a result established by Lovelock in the case of four-dimensional manifolds.