In this paper, we compute Lr (Sr) for an isometric immersion x : Mn → $\overline M^{n+1}_c$ , from an n-dimensional Riemannian manifold Mn into an (n+1)-dimensional Riemannian manifold $\overline M^{n+1}_c$ , of constant sectional curvature c. Here, by Lr we mean the linearization of the second order differential operator associated to the (r+1)-th elementary symmetric function Sr+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.