Complete hypersurfaces with Hk=0 in a unit sphere

By investigating hypersurfaces Mn in the unit sphere Sn+1(1) with Hk=0 and with two distinct principal curvatures, we give a characterization of torus S1(k/n)×Sn−1((n−k)/n). We extend recent results of T. Hasanis et al. [T. Hasanis, A. Savas-Halilaj, T. Vlachos, Complete Minimal hypersurfaces in a sphere, Monatsh. Math. 145 (2005) 301-305] and T. Otsuki [T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970) 145-173].