Homogeneous Riemannian Manifolds

Abstract

AbstractThis chapter is devoted to homogeneous spaces and homogeneous Riemannian manifolds. We discuss transitive isometry groups for a given homogeneous Riemannian manifold and topological properties of homogeneous spaces. We consider the infinitesimal structure of homogeneous Riemannian manifolds and the structure of the set of G-invariant Riemannian metrics on a homogeneous space G=H. Moreover, we derive useful formulas for the sectional curvature, the Ricci curvature, and the scalar curvature of a given homogeneous Riemannian space. Special attention is paid to Killing vector fields of constant length and the corresponding isometric flows on symmetric spaces. It is proved that such a flow on any symmetric space is free or induced by a free isometric action of the circle and consists of Clifford–Wolf translations. Various examples of the Killing vector fields of constant length, generated by the isometric effective almost free but not free actions of the circle on the Riemannian manifolds close in some sense to symmetric spaces, are constructed. Finally, we discuss various topological and algebraical restrictions for homogeneous Riemannian spaces with positive or negative sectional curvature, as well as positive or negative Ricci curvature, and the structure of compact homogeneous Riemannian spaces with Killing vectors fields of constant length.