  This paper deals with the problem of describing the vector spaces of
divergence-free, natural tensors on a pseudo-Riemannian manifold that are
second-order; i.e., that are defined using only second derivatives of the
metric.
  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 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 tensors that are
divergence-free, thus refining the original Lovelock's statement.
