2-semiparallel surfaces in space forms. I: Two particular cases (Q2718820)

Submanifolds whose fundamental form \(h\) is parallel (\(\overline{\nabla}h=0\)) with respect to the Vander Waerden-Bortolotti connection in space forms \(N^n(c)\) of constant curvature \(c\) have been classified by \textit{D. Ferus} for \(c=0\) [Math. Ann. 247, 81-93 (1980; Zbl 0446.53041)] and for \(c \neq 0\) by \textit{E. Backes} and \textit{H. Reckziegel} [Math. Ann. 263, 419-433 (1983; Zbl 0499.53045)] and \textit{M. Takeuchi} independently in [Manifolds and Lie groups, Pap. in Honor of Y. Matsushima, Prog. Math. 14, 429-447 (1981; Zbl 0481.53047)]. Semiparallel surfaces in Euclidean space and in space forms have been classified by \textit{J. Deprez} [J. Geom. 25, 192-200 (1985; Zbl 0582.53042)] and by \textit{Ü. Lumiste} [in Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, 779-864 (2000; Zbl 0964.53002)] respectively. In the present paper the authors classify the 2-semiparallel surfaces (\(\overline R\circ (\overline\nabla h)=0\)) in space forms for two cases: at first surfaces with a flat normal connection \(\nabla^\bot\) and secondly, pointwise isotropic surfaces and non-flat \(\overline{\nabla}\). NEWLINENEWLINENEWLINEReviewer's remark: For the general case see \textit{Ü. Lumiste}'s paper in Proc. Est. Acad. Sci., Phys. Math. 49, No. 4, 203-214 (2000; Zbl 1005.53045).