Compact graphs over a sphere of constant second order mean curvature

The aim of this work is to show that a compact smooth star-shaped hypersurface $\Sigma^n$ in the Euclidean sphere $\mathbb{S}^{n+1}$ whose second function of curvature $S_2$ is a positive constant must be a geodesic sphere $\mathbb{S}^{n}(\rho)$. This generalizes a result obtained by Jellett in $1853$ for surfaces $\Sigma^2$ with constant mean curvature in the Euclidean space $\mathbb{R}^3$ as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere $\mathbb{S}^{n+1}$ with constant mean curvature. In order to prove our theorem we shall present a formula for the operator $L_{r}(g)=div\left(P_r\nabla g\right)$ associated with a new support function $g$ defined over a hypersurface $M^n$ in a Riemannian space form $M_{c}^{n+1}$.