Curvature identities for Einstein manifolds of dimension 5 and 6

TL;DR

This paper derives explicit curvature identities for 5- and 6-dimensional Einstein manifolds, confirming their consistency with previous results and providing supporting examples.

Contribution

It explicitly formulates Patterson's curvature identities for 5- and 6-dimensional Einstein manifolds and verifies their equivalence with earlier derived identities.

Findings

Curvature identities explicitly formulated for 5- and 6-dimensional Einstein manifolds.

Confirmed identities are consistent with previous results.

Provided examples supporting the theorems.

Abstract

Patterson discussed the curvature identities on Riemannian manifolds in [14], and a curvature identity for any 6-dimensional Riemannian manifold was independently derived from the Chern-Gauss-Bonnet Theorem [8]. In this paper, we provide the explicit formulae of Patterson's curvature identity that holds on 5-dimensional and 6-dimensional Einstein manifolds. We confirm that the curvature identities on the Einstein manifold from the previous work [8] are the same as the curvature identities deduced from Patterson's result. We also provide examples that support the theorems.