Killing vector fields of constant length on compact hypersurfaces

We show that if a compact hypersurface $M\subset {\mathbb{R}}^{n+1}$, $n \geq3$,admitsanonzeroKillingvectorfield$X$ofconstantlengththen$n$isevenand$M$isdiffeomorphictotheunithypersphereof$$\mathbb{R}$^{n+1}$. Actually, we show that $M$ is a complex ellipsoid in $$\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}.