Dimensional curvature identities on pseudo-Riemannian geometry

Abstract

For a fixed n ∈ 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 n . In this paper, we re-elaborate recent results by Gilkey–Park–Sekigawa regarding these p -covariant 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, both by dropping a symmetry hypothesis and by including p -covariant curvature identities, for any even p . Thus, for any dimension n , our main result describes the first space (i.e., that of highest weight) of p -covariant dimensional curvature identities, for any even p .