M-eigenvalues of Riemann Curvature Tensor of Conformally Flat Manifolds

TL;DR

This paper extends the understanding of M-eigenvalues of Riemann curvature tensors to higher-dimensional conformally flat manifolds, providing explicit expressions, invariance properties, and applications to spacetime models.

Contribution

It generalizes previous results to higher dimensions, derives explicit formulas, and explores invariance and applications in general relativity.

Findings

M-eigenvalues uniquely determine the Riemann curvature tensor.

Derived explicit expressions for M-eigenvalues and eigenvectors.

Applied the theory to compute M-eigenvalues of de Sitter spacetime.

Abstract

We generalized Xiang, Qi and Wei's results on the M-eigenvalues of Riemann curvature tensor to higher dimensional conformal flat manifolds. The expression of M-eigenvalues and M-eigenvectors are found in our 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 discussed the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We proved that the M-eigenvalue can determine the Riemann curvature tensor uniquely and generalize the real M-eigenvalue to complex cases. In the last part of our paper, we give an example to compute the M-eigentriple of de Sitter spacetime which is well-known in general relativity.