Abstract.Let M" denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic n + 1 space or in an open hemisphere of S"+ '. We show that if two consecutive mean curvatures of M are constant then M is in fact a geodesic sphere. The proof uses the generalized Minkowski integral formulas for a hypersurface of a complete simply connected space form. These Minkowski formulas are derived from an integral formula for submanifolds in which the ambient Riemannian manifold M possesses a generalized position vector field; that is a vector field Y whose covariant derivative is at each point a multiple of the identity. In addition we prove that if M is complete and connected with the covariant derivative of Y exactly the identity at each point then M is isometric to Euclidean space.

No abstract

Abstract.Let M" denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic n + 1 space or in an open hemisphere of S"+ '. We show that if two consecutive mean curvatures of M are constant then M is in fact a geodesic sphere. The proof uses the generalized Minkowski integral formulas for a hypersurface of a complete simply connected space form. These Minkowski formulas are derived from an integral formula for submanifolds in which the ambient Riemannian manifold M possesses a generalized position vector field; that is a vector field Y whose covariant derivative is at each point a multiple of the identity. In addition we prove that if M is complete and connected with the covariant derivative of Y exactly the identity at each point then M is isometric to Euclidean space.

No abstract

Given tensors A and B of type (k, k) on a Riemannian manifold M we construct in a natural way a 2k form Fk(A, B). If A and B satisfy the generalized Codazzi equations then this 2k form is closed. In particular if R2k denotes the 2fcth curvature operator then F2*(^2*> ^2*) >* (UP to a constant multiple) the fcth Pontrjagin class of M. By means of a theorem of Gilkey we give conditions sufficient to guarantee that a form constructed from more complicated expressions involving the curvature operators does in fact belong to the Pontrjagin algebra. As a corollary we obtain Thorpe's vanishing theorem for manifolds with constant 2/>th sectional curvature.