Semi-parallel hypersurfaces of a real space form
Let \(M\) be a hypersurface of an \((n+1)\)-dimensional standard space \(N\) of constant curvature \(c\). \(M\) is said to be semi-parallel, iff the second fundamental form \(h: TM\times_ M TM\to\mathbb{R}\) of \(M\) satisfies \(h(R(X,Y)U,V)+h(U,R(X,Y)V)=0\), where \(R\) denotes the curvature tensor of \(M\). The author shows that in case \(c\neq 0\) there are three (and no other) possibilities for semi-parallel hypersurfaces \(M\): (1) \(n=2\) and \(M\) is flat, (2) \(M\) is parallel, and (3) \(M\) is a rotation hypersurface whose radius function depends linearly on the arc length parameter of the profile curve. This classification follows from the fact that \(M\) has at most two principal curvatures \(\lambda\) and \(\mu\) everywhere, which satisfy \(\lambda\cdot\mu+c=0\) in case \(\lambda\neq\mu\). Semi-parallel hypersurfaces in Euclidean space were already classified by \textit{J. Deprez} [Rend. Semin. Mat., Torino 44, 303-316 (1986; Zbl 0616.53018)].
scientific article; zbMATH DE number 78182
scientific article