On hypersurfaces into Riemannian spaces of constant sectional curvature

Abstract

In this paper, we compute L_r(S_r) for an isometric immersion x : Mⁿ → M_cⁿ⁺¹, from an n-dimensional Riemannian manifold Mⁿ into an (n+1)-dimensional Riemannian manifold M_cⁿ⁺¹, of constant sectional curvature c. Here, by L_r we mean the linearization of the second order differential operator associated to the (r+1)-th elementary symmetric function S_r+1 on the eigenvalues of the second fundamental form A of x. The resulting formulae are then applied to study how the behavior of higher-order mean curvature functions of Mⁿ influence its geometry.