Homogeneity for a Class of Riemannian Quotient Manifolds

TL;DR

This paper investigates the conditions under which quotients of certain homogeneous Riemannian manifolds remain homogeneous, extending classical results and characterizing isometries of constant displacement.

Contribution

It proves the Homogeneity Conjecture for a class of Riemannian quotient manifolds when the base space is well-understood, and extends the classification of isometries to these settings.

Findings

Homogeneity of quotients characterized by isometries of constant displacement.

Full isometry groups of certain homogeneous spaces determined.

Extensions to pseudo-Riemannian cases discussed.

Abstract

We study riemannian coverings $\varphi: \widetilde{M} \to \Gamma\backslash \widetilde{M}$ where $\widetilde{M}$ is a normal homogeneous space $G/K_1$ fibered over another normal homogeneous space $M = G/K$ and $K$ is locally isomorphic to a nontrivial product $K_1\times K_2$. The most familiar such fibrations $\pi: \widetilde{M} \to M$ are the natural fibrations of Stieffel manifolds $SO(n_1 + n_2)/SO(n_1)$ over Grassmann manifolds $SO(n_1 + n_2)/[SO(n_1)\times SO(n_2)]$ and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings $\varphi: \widetilde{M} \to \Gamma\backslash \widetilde{M}$ are the universal riemannian coverings of spherical space forms. When $M = G/K$ is reasonably well understood, in particular when $G/K$ is a riemannian symmetric space or when $K$ is a connected subgroup of maximal rank in $G$, we show that the Homogeneity Conjecture holds for $\widetilde{M}$. In other words we show that $\Gamma\backslash \widetilde{M}$ is homogeneous if and only if every $\gamma \in \Gamma$ is an isometry of constant displacement. In order to find all the isometries of constant displacement on $\widetilde{M}$ we work out the full isometry group of $\widetilde{M}$, extending Elie Cartan's determination of the full group of isometries of a riemannian symmetric space. We also discuss some pseudo-riemannian extensions of our results.