Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8898370 | Differential Geometry and its Applications | 2018 | 18 Pages |
Abstract
We study Riemannian coverings Ï:MËâÎ\MË where MË 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 K1ÃK2. The most familiar such fibrations Ï:MËâM are the natural fibrations of Stiefel manifolds SO(n1+n2)/SO(n1) over Grassmann manifolds SO(n1+n2)/[SO(n1)ÃSO(n2)] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings Ï:MËâÎ\MË 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 MË. In other words we show that Î\MË is homogeneous if and only if every γâÎ is an isometry of constant displacement. In order to find all the isometries of constant displacement on MË we work out the full isometry group of MË, extending Ãlie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Joseph A. Wolf,