2016/09/19 by Wolf, Joseph A.
#22F30 #53C20 #53C26 #53C35 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1609.05576
We study riemannian coverings φ: \widetildeM → Γ\backslash \widetildeM where \widetildeM 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: \widetildeM → M are the natural fibrations of Stieffel 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: \widetildeM → Γ\backslash \widetildeM 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 \widetildeM. In other words we show that Γ\backslash \widetildeM 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 \widetildeM we work out the full isometry group of \widetildeM, 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.