Homogeneity for a Class of Riemannian Quotient Manifolds

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.