Patterson discussed the curvature identities on Riemannian manifolds based on the skew-symmetric properties of the generalized Kronecker delta, and a curvature identity for any 6-dimensional Riemannian manifold was independently derived from the Chern-Gauss-Bonnet Theorem. 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 derived from the Chern-Gauss-Bonnet Theorem are the same as the curvature identities deduced from Patterson's result. We also provide examples that support the theorems.
EUH, YUNHEE; KIM, JIHUN; and PARK, JEONGHYEONG
"Curvature identities for Einstein manifolds of dimensions 5 and 6,"
Turkish Journal of Mathematics: Vol. 46:
6, Article 33.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss6/33