$M$-eigenvalues of the Riemann curvature tensor

Abstract

The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvature tensor is introduced from the minimization of an associated function. The M-eigenvalues of the Riemann curvature tensor always exist and are real. They are invariants of the Riemann curvature tensor. The associated function of the Riemann curvature tensor is always positive at a point if and only if the M-eigenvalues of the Riemann curvature tensor are all positive at that point. We investigate the M-eigenvalues for the simple cases, such as the 2D case, the 3D case, the constant curvature and the Schwarzschild solution, and all the calculated M-eigenvalues are related to the curvature invariants.