A sufficient condition for congruency of orbits of Lie groups and some applications

TL;DR

This paper establishes a sufficient condition for when all orbits of a Lie group action are congruent, providing new examples and simplified proofs for known results in symmetric spaces.

Contribution

It introduces a new sufficient condition for orbit congruency in Lie group actions and applies it to derive both existing and novel results.

Findings

Provided a unified proof for known congruence results.

Identified new examples of isometric actions with congruent orbits.

Simplified understanding of orbit congruency in symmetric spaces.

Abstract

We give a sufficient condition for isometric actions to have the congruency of orbits, that is, all orbits are isometrically congruent to each other. As applications, we give simple and unified proofs for some known congruence results, and also provide new examples of isometric actions on symmetric spaces of noncompact type which have the congruency of orbits.