The metric geometry of infinite dimensional Lie groups and their homogeneous spaces

TL;DR

This paper explores the geometry of infinite-dimensional Lie groups with Finsler metrics and their homogeneous spaces, analyzing geodesic structures and applications to operator groups and mapping groups.

Contribution

It systematically studies the metric and geodesic structures of infinite-dimensional Lie groups and their homogeneous spaces, focusing on bi-invariant metrics and their shortest paths.

Findings

One-parameter groups are shortest paths in groups with bi-invariant metrics.

Characterization of all shortest paths beyond one-parameter groups.

Applications to Banach space operator manifolds and groups of maps.

Abstract

We study the geometry of Lie groups $G$ with a continuous Finsler metric, assuming the existence of a subgroup $K$ such that the metric is right-invariant for the action of $K$. We present a systematic study of the metric and geodesic structure of homogeneous spaces $M$ obtained by the quotient $M\simeq G/K$. Of particular interest are left-invariant metrics of $G$ which are then bi-invariant for the action of $K$. We then focus on the geodesic structure of groups $K$ that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.