Minimal Integrity Bases of Invariants of Second Order Tensors in a Flat Riemannian Space

TL;DR

This paper investigates the invariants of second order tensors in flat Riemannian spaces, establishing minimal integrity bases for symmetric and antisymmetric tensors, with applications in physics like electrodynamics.

Contribution

It introduces minimal integrity bases for second order symmetric and antisymmetric tensors in flat Riemannian spaces, including special cases and applications.

Findings

Coefficients of characteristic polynomials are real polynomial invariants.

Minimal integrity bases for symmetric and antisymmetric tensors are provided.

Applications discussed in Minkowski space and electrodynamics.

Abstract

In this paper, we study invariants of second order tensors in an $n$-dimensional flat Riemannian space. We define eigenvalues, eigenvectors and characteristic polynomials for second order tensors in such an $n$-dimensional Riemannian space and show that the coefficients of the characteristic polynomials are real polynomial invariants of that tensor. Then we give minimal integrity bases for second order symmetric and antisymmetric tensors, respectively, and study their special cases in the Minkowski space and applications in electrodynamics, etc.