  The curvature tensor of a pseudo-Riemannian metric, and its covariant
derivatives, satisfy certain identities that hold on any manifold of dimension
less or equal than $n$.
  In this paper, we re-elaborate recent results by Gilkey-Park-Sekigawa
regarding $p$-covariant dimensional curvature identities, for $p=0,2$. To this
end, we use the classical theory of natural operations, that allows us to
simplify some arguments and to generalize the description of
Gilkey-Park-Sekigawa.
  Thus, our main result describes the first space of $p$-covariant dimensional
curvature identities, for any even $p$.
