Killing vector fields of constant length on compact hypersurfaces

Abstract

We show that if a compact hypersurface $M\subset {\mathbb{R}}^{n+1}$, $n\ge 3$, 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}.