Homogeneity for a class of Riemannian quotient manifolds

Abstract

We study Riemannian coverings φ:M˜→Γ\\M˜ where M˜ is a normal homogeneous space G/K1 fibered over another normal homogeneous space M=G/K and K is locally isomorphic to a nontrivial product K1×K2. The most familiar such fibrations π:M˜→M are the natural fibrations of Stiefel manifolds SO(n1+n2)/SO(n1) over Grassmann manifolds SO(n1+n2)/[SO(n1)×SO(n2)] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings φ:M˜→Γ\\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 M˜. In other words we show that Γ\\M˜ is homogeneous if and only if every γ∈Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M˜ we work out the full isometry group of M˜, extending Élie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.