A characterization of quadric constant scalar curvature hypersurfaces of spheres (Q472093)

The authors prove the following result (Theorem 1.1) : Let \(M \subset S^4\) be a complete orientable hypersurface with constant scalar curvature \(R\), and let \(l_v, f_v :M\rightarrow \mathbb{R}\) be the real functions defined by \(l_v(x)=\langle x,v\rangle, f_v(x)=\langle \nu (x),v\rangle\), where \(\nu :M\rightarrow S^4\) is a Gauss map of \(M\). If \(l_v=\lambda f_v\) for some nonzero vector \(v \in \mathbb{R}^5\) and some real number \(\lambda\), then \(M\) is either totally umbilical (a Euclidean sphere) or \(M\) is a Cartesian product of Euclidean spheres. In Section 4 of the paper, an example illustrates that the completeness condition is necessary.