The \(r\)-mean curvature and rigidity of compact hypersurfaces in the Euclidean space (Q824400)

Denote by \(M^n\) an orientable Riemannian manifold in Euclidean space \(\mathbb R^{n+1}\) and by \(x:M^n\rightarrow \mathbb R^{n+1}\) an isometric immersion. Consider a globally defined unit normal vector field \(N\) and the second fundamental form \(A\) of the hypersurface with respect to \(N\). Then the \(r\)-mean curvature of \(M^n\) is \begin{center} \(H_r={\binom{n}{r}}^{-1}S_r,\) \end{center} where \(S_r\) is the \(r\)-elementary symmetric function of the eigenvalues of \(A\), for \(r = 1,\dots, n\), and \(S_0 = 1\). The present paper gives characterizations of the round sphere among closed hypersurfaces under some suitable conditions on \(H_r\).