Spectral properties of Killing vector fields of constant length

Abstract

This paper is devoted to the study of properties of Killing vector fields of constant length on Riemannian manifolds. If $\mathfrak{g}$ is a Lie algebra of Killing vector fields on a given Riemannian manifold $(M,g)$, and $X\in \mathfrak{g}$ has constant length on $(M,g)$, then we prove that the linear operator $\mathrm{ad}(X):\mathfrak{g}\to \mathfrak{g}$ has a pure imaginary spectrum. More detailed structure results on the corresponding operator $\mathrm{ad}(X)$ are obtained. Some special examples of vector fields of constant length are constructed.

10 pages, comments are welcome