Hypersurfaces of restricted type in Minkowski space
A submanifold \(M^n_r\), \(n\) being the dimension and \(r\) the index, of Minkowski space \(E^m_1\) is said to be of restricted type if its shape operator, with respect to the mean curvature vector, is the restriction of a fixed linear transformation of \(E^m_1\) to the tangent space of \(M^n_r\) at every point of \(M^n_r\). The study of this kind of submanifolds was initiated by \textit{B. Y. Chen, F. Dillen, L. Verstraelen} and \textit{L. Vranken} [J. Geom. 46, 20-32 (1993; Zbl 0778.53002)] where they stated the problem of classifying all submanifolds of restricted type of pseudo-Euclidean spaces. As for the classification of hypersurfaces of restricted type in \(E^{n+1}_1\), the authors show that it suffices to establish which of the parallel hypersurfaces of \(E^{n+1}_1\) are of restricted type. To do that the authors deal with the minimal of the shape operator, showing that the hypersurface has to be isoparametric. Then, following the papers by \textit{H. B. Lawson} [Ann. Math. 89, 187-197 (1969; Zbl 0174.24901)] and \textit{M. A. Magid} [Tsukuba J. Math. 8, 31-54 (1984; Zbl 0549.53052)], they obtain the main result: The only hypersurfaces of restricted type of \(E^{n+1}_1\) are (i) the minimal ones; (ii) an open part of \(S^k \times E^{n-k}_1\), \(S^k_1 \times E^{n-k}\), \(H^k \times E^{n-k}\), \(S^k_1\) and \(H^n\), with \(1\leq k\leq n-1\); and (iii) an open part of a cylinder on a plane curve of restricted type.
classification of hypersurfaces of restricted type
scientific article; zbMATH DE number 929921