A curvature identity on a~6-dimensional Riemannian manifold and its applications

Abstract

We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. We also introduce some applications of this curvature identity.2000 Mathematics Subject Classification. 53B20, 53C25.

Curvature identities for Einstein manifolds of dimension 5 and 6

“…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.…”

“…In [8], the authors consider the one-parameter deformation g(t) of g, using the fact that Euler characteristic is a topological invariant for the deformation, they got the universal curvature identity which holds on any 6-dimensional Riemannian manifold. In particular, they have the following.…”

“…We derive the curvature identities from Patterson's curvature identity. In Section 2, we recall the previous results for a 6-dimensional Riemannian manifold [8] and introduce Patterson's curvature identity. In Section 3 and Section 4, we prove Theorem A and Theorem B, respectively.…”

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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.

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Characterizing the harmonic manifolds by the eigenfunctions of the Laplacian

“…But Damek and Ricci [3] found that there are many counterexamples if dimension n ≥ 7. Euh, Park and Sekigawa [4] provide a new proof of the Lichnerowicz conjecture for dimension n = 4, 5 in a slightly more general setting using universal curvature identities.…”