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

Curvature identities for Einstein manifolds of dimension 5 and 6