Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by f_ij{phi_{ij}} the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of f_ij{phi_{ij}} . We obtain two integral formulas by using Φ and the polynomial P_H,m(x)=x²+ fracn(n-2m)?{nm(n-m)}H x -n(1+H²){P_{H,m}(x)=x^{2}+ frac{n(n-2m)}{sqrt{nm(n-m)}}H x -n(1+H^{2})} . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either F = B_H,m{Phi=B_{H,m}} or F = B_H,n-m{Phi=B_{H,n-m}} . In particular, M ⁿ is the hypersurface S^n-m(r)×S^m(?{1-r²}){S^{n-m}(r)times S^{m}(sqrt{1-r^{2}})} .