Let $\phi:M\to\mathbb{ S}^{ n+1}\subset\mathbb{ R}^{ n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{ R}^{ n+2}$, let us denote by $\ell_v:M\to\mathbb{ R}$ the function given by $\ell_v(x)=\phi(x),v$ and by $f_v:M\to\mathbb{ R}$, the function given by $f_v(x)=\nu(x),v$, where $\nu:M\to\mathbb{ S}^{ n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\ne{ \bf 0}$ and some real number $\lambda$, we have that $\ell_v=\lambda f_v$, then, $\phi(M)$ is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface $M^n$ in $\mathbb{ S}^{ n+1}$ which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to $2n+4$.
Categories: math.DG