On Clifford-Wolf homogeneous Riemannian manifolds

Abstract

In this chapter, using connections between Clifford–Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford–Wolf homogeneous Riemannian manifolds.We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J. E. D’Atri and H. K. Nickerson. The next goal of the chapter is to study properties of Clifford–Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres, and discuss numerous connections between these spaces and various classical objects. Finally, we consider some results related to restrictively Clifford–Wolf homogeneous Riemannian manifolds.