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# Communications in Mathematical Sciences

## Volume 16 (2018)

### Number 8

### $M$-eigenvalues of the Riemann curvature tensor

Pages: 2301 – 2315

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n8.a10

#### Authors

#### 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.

#### Keywords

curvature tensor, Riemann tensor, Ricci scalar, eigenproblem, $M$-eigenvalue, Schwarzschild solution, general relativity, invariants

#### 2010 Mathematics Subject Classification

15A18, 15A69, 53Z05, 83C99

Received 25 March 2018

Accepted 23 August 2018

Published 18 April 2019