Dimensional curvature identities on pseudo-Riemannian geometry

Abstract

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 $p$.

Polished version. 15 pages

References in corpus (6)

• The Riemann-Lovelock Curvature Tensor
• On the naturalness of Einstein's equation
• Lovelock's theorem revisited
• Universal curvature identities II
• Universal curvature identities
• Universal curvature identities and Euler Lagrange Formulas for Kaehler manifolds

Cited by in corpus (2)

• A remark on the invariant theory of real Lie groups
• On the uniqueness of the Gauss-Bonnet-Chern formula (after Gilkey-Park-Sekigawa)