On hypersurfaces in a conformally flat Riemannian manifold with parallel second fundamental form

Abstract:

      In this paper, we prove the followingTheorem. Let Cⁿ⁺¹ ( n >5 ) be a conformally flat Riemannian anifola of dimension n + 1 . If Mn is a hypersurface immersed isometrically in Cⁿ⁺¹ over which the second fundamental form is covariant constant, then there are three posible cases only:I . locally Mⁿ= S^p×S^q×S^r, p+q+r=n;Ⅱ . locally Mⁿ=S^p×S^q, p + q = n , where S^k is k- dimensional Riemannian space of constant curvature;III. Mⁿ is umbilical and conformally flat. Moreover, if Mⁿ is connected and complete, then the result holds globally.