# Spectral properties of Killing vector fields of constant length

## TL;DR

This paper investigates the spectral properties of Killing vector fields of constant length on Riemannian manifolds, revealing that the adjoint operator associated with such fields has a purely imaginary spectrum and providing structural insights and examples.

## Contribution

It establishes that the adjoint operator of a constant-length Killing vector field has a purely imaginary spectrum and offers detailed structural results and examples.

## Key findings

- The operator ad(X) has a purely imaginary spectrum.
- Structural properties of ad(X) are characterized.
- Examples of constant-length Killing vector fields are constructed.

## 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 $\operatorname{ad}(X):\mathfrak{g} \rightarrow \mathfrak{g}$ has a pure imaginary spectrum. More detailed structure results on the corresponding operator $\operatorname{ad}(X)$ are obtained. Some special examples of vector fields of constant length are constructed.