M-eigenvalues of the Riemann curvature tensor

Hua Xiang, Liqun Qi, Yimin Wei

Research output: Journal article publicationJournal articleAcademic researchpeer-review

5 Citations (Scopus)

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 Meigenproblem 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.

Original languageEnglish
Pages (from-to)2301-2315
Number of pages15
JournalCommunications in Mathematical Sciences
Volume16
Issue number8
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Curvature tensor
  • Eigenproblem
  • General relativity
  • Invariants
  • M-eigenvalue
  • Ricci scalar
  • Riemann tensor
  • Schwarzschild solution

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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