A characterization of quadric constant mean curvature hypersurfaces of spheres

Let phi : M -> Sn+1 subset of Rn+2 be an immersion of a complete n-dimensional oriented manifold. For any upsilon is an element of Rn+2, let us denote by l(upsilon) : M -> R the function given by l(upsilon)(x) = [phi(x), upsilon] and by f(upsilon) : M -> R, the function given by f(upsilon)(x) = [nu(x), upsilon], where. : M -> Sn+1 subset of Rn+2 is a Gauss map. We will prove that if M has constant mean curvature, and, for some upsilon not equal 0 and some real number lambda, we have that l(upsilon) = lambda f(upsilon), 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 Sn+1 which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4.