Hypersurfaces with Constant k-th Mean Curvature in a Unit Sphere and Euclidean Space

Let M^n be an n(n ≥ 3)-dimensional complete connected and oriented hypersurface in a unit sphere S^n+1(1) or Euclidean space R^n+1 with constant k-th mean curvature Hk 0(k n) and with two distinct principal curvatures ƛ and μ such that the multiplicity of ƛ is n-1. We show that (1) in the case of S^n+1(1), if k ≥ 3 and │h│^2 ≤ (n-1)t2^2/k=k 2 +t2^-2/k , then M^n is isometric to the Riemannian product S^1( p√ 1-a^2)*S^n-1(a), where t2 is the positive real root of the function PHk (t) = kt^ k-2/ k -(n-k)t +nHk; (2) in the case of R^n+1, if │h│^≤ (n-1)(nHk/(n-k))^2/k , then Mn is isometric to the Riemannian product S^n-1(a)*R. We extend some recent results to the case k ≥ 3.