Invariant proper metrics on coset spaces

TL;DR

This paper investigates conditions under which coset spaces G/H admit proper G-invariant metrics that induce their topology, extending known results from groups to coset spaces.

Contribution

It establishes necessary and sufficient conditions for the existence of proper G-invariant metrics on coset spaces G/H.

Findings

Identifies criteria for proper G-invariant metrics on G/H

Extends known metric existence results from groups to coset spaces

Provides a framework for analyzing metric structures on homogeneous spaces

Abstract

It is known that for every second countable locally compact group G, there exists a proper G-invariant metric which induces the topology of the group. This is no longer true for coset spaces G/H viewed as G-spaces. We study necessary and sufficient conditions which ensure the existence of such metrics on G/H.