Bounded Isometries and Homogeneous Quotients

TL;DR

This paper characterizes bounded isometries in certain homogeneous metric spaces, including nilmanifolds, and provides evidence for a conjecture relating homogeneity of quotients to constant displacement isometries.

Contribution

It explicitly describes bounded displacement isometries in a broad class of spaces and offers an alternative proof of a classical result, supporting a longstanding conjecture.

Findings

Bounded isometries are of constant displacement.

Provides an explicit description of bounded isometries in homogeneous spaces.

Supports the conjecture linking homogeneity of quotients to constant displacement isometries.

Abstract

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author's 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if $\Gamma \backslash M$ is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold $M$, then $\Gamma \backslash M$ is homogeneous if and only if each isometry $\gamma \in \Gamma$ is of constant displacement. Our description of bounded isometries gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford's use of quaternions in non--euclidean geometry, and we sketch that in a historical note.