Closed hypersurfaces of \(S^4\) with two constant symmetric curvatures
This paper deals with closed hypersurfaces of the 4-dimensional Euclidean sphere \(S^4\) with given geometric properties. The second fundamental form of this kind of immersions is determined by three symmetric curvatures: the mean curvature \(H\), the scalar curvature \(k\) and the Gauss-Kronecker curvature \(K\). A hypersurface \(M\) is said to be isoparametric if all of them are constant. Combining results of the authors [\textit{S. C. de Almeida} and \textit{F. G. B. Brito}, Duke Math. J. 61, 195-206 (1990; Zbl 0721.53056)] and \textit{S.-P. Chang} [Commun. Anal. Geom. 1, 71-100 (1993; Zbl 0791.53058)], a closed hypersurface in \(S^4\) with constant mean curvature and constant scalar curvature is isoparametric. In this paper, the authors prove in their first theorem that a closed oriented hypersurface in \(S^4\) with constant Gauss-Kronecker curvature and constant nonnegative scalar curvature is isoparametric; in their second theorem, they suppose that \(H\) and \(K\neq 0\) are constant and, in addition, \(-3 \leq H/K\), and obtain the same conclusion (\(M\) is isoparametric).
isoparametric hypersurfaces of the 4-sphere
scientific article; zbMATH DE number 1139492