Killing vector fields of constant length on compact hypersurfaces

TL;DR

This paper characterizes compact hypersurfaces in Euclidean space that admit non-zero Killing vector fields of constant length, showing they are essentially complex ellipsoids and providing a simplified proof of a recent related theorem.

Contribution

It proves that such hypersurfaces must be even-dimensional spheres or complex ellipsoids, offering a new classification result and a simpler proof of an existing theorem.

Findings

Hypersurfaces with constant-length Killing fields are complex ellipsoids.

Such hypersurfaces are diffeomorphic to even-dimensional spheres.

The paper provides a simplified proof of a recent theorem by Deshmukh.

Abstract

We show that if a compact hypersurface $M \subset \mathbb{R}^{n+1}$, $n \geq3$, admits a non zero Killing vector field $X$ of constant length then $n$ is even and $M$ is diffeomorphic to the unit hypersphere of $\mathbb{R}^{n+1}$. Actually, we show that $M$ is a complex ellipsoid in $\mathbb{C}^{N} = \mathbb{R}^{n+1}$. As an application we give a simpler proof of a recent theorem due to S. Deshmukh \cite{De12}.