Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

Abstract

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$\n whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H,\xa0g), such that G is one of the compact classical Lie groups $${{\\,\\mathrm{SO}\\,}}(n)$$\n , $$\\mathrm{U}(n)$$\n , and H is a diagonally embedded product $$H_1\\times \\cdots \\times H_s$$\n , where $$H_j$$\n is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H,\xa0g) with H semisimple.