Compact geodesic orbit manifolds with regular stabilizer

Abstract

We study geodesic orbit Riemannian manifolds, namely manifolds where all geodesics are orbits of one - parameter groups of isometries. Any connected such manifold is homogeneous, while the structure and existence of geodesic orbit metrics on a homogeneous space G/H largely depends on the embedding of the stabilizer subgroup H in G. In this article, we study and characterize the geodesic orbit metrics in the extensive class of spaces G/H with G compact and H a semisimple regular subgroup of G (in the sense that the Lie algebra of H is normalized by a Cartan subalgebra of the Lie algebra of G). This work is a contribution to the ongoing study of compact geodesic orbit manifolds, focusing on those with semisimple stabilizer.