M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds

Abstract

We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold. The expressions of M-eigenvalues and M-eigenvectors are presented in this paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the conformal the invariance of M-eigentriple has been found. We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely. We also give an example to compute the M-eigentriple of de Sitter spacetime which is well-known in general relativity.