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Semi-parallel hypersurfaces. (English) Zbl 0616.53018

A submanifold of \(E^{n+1}\) is parallel (extrinsically locally symmetric), if its second fundamental form is parallel, \(D\alpha =0\). It is semiparallel, if the curvature tensor acts trivially on it, \(R\cdot \alpha =0\). This is an extrinsic analog of the semi-symmetric spaces studied in great detail by Z. T. Szabó [J. Differ. Geom. 17, 531- 582 (1982; Zbl 0508.53025)]. The author shows that locally semi-parallel hypersurfaces have type number \(\leq 1\), or else they are spheres, cones or cylinders over such. He also shows that the Weyl tensor acts trivially on the second fundamental tensor of a hypersurface if and only if it is conformally flat. The proofs are based on Szabó’s rather intricate results, but a simpler direct proof seems possible.
Reviewer: D.Ferus

MSC:

53B25 Local submanifolds

Citations:

Zbl 0508.53025
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