Clifford-Wolf homogeneous Riemannian manifolds

TL;DR

This paper classifies simply connected Clifford-Wolf homogeneous Riemannian manifolds using Killing vector fields and explores properties of Clifford-Killing spaces on spheres, connecting them to classical geometric objects.

Contribution

It provides a comprehensive classification of Clifford-Wolf homogeneous manifolds and analyzes Clifford-Killing spaces on spheres, linking them to classical geometry.

Findings

Classification of simply connected Clifford-Wolf homogeneous manifolds

Characterization of Clifford-Killing spaces on odd-dimensional spheres

Connections between Killing vector fields and classical geometric objects

Abstract

In this paper, 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. In the last part of the paper we 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.