Rigidity of hypersurfaces with constant higher order mean curvature in space forms

We consider complete oriented hypersurfaces M immersed into a Riemannian space form with constant higher order mean curvature and having two principal curvatures, one of them simple. We use the fact that these hypersurfaces are in fact rotation hypersurfaces and impose further restrictions on the Ricci curvature to prove that they be must be isoparametric, therefore generalizing or giving alternate proofs of other characterization results.