Killing vector fields of constant length on Riemannian normal homogeneous spaces (Q326369)

Riemannian normal homogeneous space are investigated. These spaces are a special kind of naturally reductive homogeneous spaces. They may be considered as a generalization of Riemannian symmetric spaces, including the non-compact type. A notion of Killing vector field of constant length is used. The authors name them as Clifford-Killing vector fields or CK vector fields. These correspond to isometries of constant displacement. The main result of this article is the following. Let \(G\) be a compact connected simple Lie group and \(H\) a closed subgroup with \(0 < \dim H<\dim G\). Fix a normal Riemannian metric on the homogeneous space \(M = G/H\). Suppose that there is a nonzero vector \(v\) in the Lie algebra of \(G\), which defines a CK vector field on \(M\). Then \(M\) is a complete locally symmetric Riemannian manifold, and its universal Riemannian cover is an odd-dimensional sphere of constant curvature or a Riemannian symmetric space \(\mathrm{SU}(2n)/\mathrm{Sp}(n)\). The proof uses a consideration of all classes of simple Lie groups -- both classical and exceptional ones. This result shows that the existence of nontrivial CK vector fields gives very strong restrictions on a Riemannian homogeneous space, at least when such space is Riemannian normal homogeneous. When a Lie group \(G\) is compact but not simple, a Riemannian normal homogeneous space \(M\) does not have in general a perfect local decomposition into symmetric spaces. So the study of CK vector fields in this situation is still open.