Covariants,joint invariants and the problem of equivalence in the invariant theory of Killing tensors defined in pseudo-Riemannian spaces of constant curvature

TL;DR

This paper extends the invariant theory of Killing tensors by introducing covariants and joint invariants, enabling classification of orthogonal coordinate webs in pseudo-Riemannian spaces of constant curvature.

Contribution

It introduces covariants and joint invariants for Killing tensors, providing new tools for classification in pseudo-Riemannian spaces.

Findings

Classified orthogonal coordinate webs in Euclidean and Minkowski planes.

Extended invariant theory to include covariants and joint invariants.

Provided illustrative examples demonstrating the classification method.

Abstract

The invariant theory of Killing tensors (ITKT) is extended by introducing the new concepts of covariants and joint invariants of (product) vector spaces of Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The covariants are employed to solve the problem of classification of the orthogonal coordinate webs generated by non-trivial Killing tensors of valence two defined in the Euclidean and Minkowski planes. Illustrative examples are provided.