Locally compact homogeneous spaces with inner metric

TL;DR

This paper reviews the structure of locally compact homogeneous spaces with inner metrics, introduces a new metric for their classification, and characterizes Carnot groups and Finslerian manifolds within this framework.

Contribution

It establishes a complete metric space structure on these spaces, characterizes Carnot groups with sub-Finslerian metrics, and describes conditions for Finslerian invariance.

Findings

The space of such homogeneous spaces is complete under the new metric.

Convergence in this space aligns with Gromov-Hausdorff convergence.

Homogeneous manifolds with inner metrics are densely represented by Lie groups with Finslerian metrics.

Abstract

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class $\Omega$ of all locally compact homogeneous spaces with inner metric is supplied with some metric $d_{BGH}$ such that 1) $(\Omega,d_{BGH})$ is a complete metric space; 2) a sequences in $(\Omega,d_{BGH})$ is converging if and only if it is converging in Gromov-Hausdorff sense; 3) the subclasses $\mathfrak{M}$ of homogeneous manifolds with inner metric and $\mathfrak{LG}$ of connected Lie groups with left-invariant Finslerian metric are everywhere dense in $(\Omega,d_{BGH}).$ It is given a metric characterization of Carnot groups with left-invariant sub-Finslerian metric. At the end are described homogeneous manifolds such that any invariant inner metric on any of them is Finslerian.