Dimensional curvature identities on pseudo-Riemannian geometry

For a fixed n∈Nn∈N, the curvature tensor of a pseudo-Riemannian metric, as well as its covariant derivatives, satisfies certain identities that hold on any manifold of dimension less than or equal to nn.In this paper, we re-elaborate recent results by Gilkey–Park–Sekigawa regarding these pp-covariant curvature identities, for p=0,2p=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, both by dropping a symmetry hypothesis and by including pp-covariant curvature identities, for any even pp.Thus, for any dimension nn, our main result describes the first space (i.e., that of highest weight) of pp-covariant dimensional curvature identities, for any even pp.