Minimal hypersurfaces of \(S^ 4\) with constant Gauss-Kronecker curvature (Q1081852)

Closed minimal hypersurfaces \(M^ 3\) with constant Gauss-Kronecker curvature and nowhere vanishing second fundamental form in the unit sphere \(S^ 4\) are classified in this paper. Precisely let K be the Gauss-Kronecker curvature of \(M^ 3\) (product of the principal curvatures). If \(K\neq 0\) then the hypersurface M is a standard minimal 3- dimensional Clifford torus in \(S^ 4\). If \(K=0\) and there are no points in M where the second fundamental form vanishes then M is the tube of a certain minimal surface \(\Sigma ^ 2\) in \(S^ 4\).