Integral formulas and hyperspheres in a simply connected space form

Abstract:

Let ${M^n}$ denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic $n + 1$ space or in an open hemisphere of ${S^{n + 1}}$. 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 $\overline M$ 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 $\overline M$ is complete and connected with the covariant derivative of $Y$ exactly the identity at each point then $\overline M$ is isometric to Euclidean space.