2-semiparallel surfaces in space forms. II: The general case (Q2718824)

A surface in \(n\) dimensional space form is said to be 2-semiparallel if (\(\overline R\circ (\overline\nabla h)=0\)). The author extended the study of \textit{K. Arslan} et al. [ Proc. Est. Acad. Sci., Phys. Math. 49, 139-148 (2000; Zbl 1005.53044)] and classified all 2-semiparallel surfaces, (Theorem 6): A surface \(M^{2}\) in a space form \(N^n(c)\) is 2-semiparallel if and only if it belongs to one of the following three mutually exclusive classes consisting of: NEWLINENEWLINENEWLINEsurfaces with surfaces with a flat Vander Waerden-Bortolotti connection \(\overline{\nabla}\), NEWLINENEWLINENEWLINEparallel surfaces with non-flat \(\overline{\nabla}\), i.e. totally umbilical surfaces and veronese surfaces, NEWLINENEWLINENEWLINElocally Euclidean surfaces (i.e. with flat \(\nabla\), or equivalent with vanishing Gaussian curvature), whose normal connection \(\nabla^\bot\) is non-flat and span\(\{{h_{ijk}}\}\) at an arbitrary point \(x \in M\) is orthogonal to the plane of normal curvature ellipse at this point; these ellipses at two arbitrary points of such an \(M^{2}\) are congruent.