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A quasi-umbilical hypersurface of a conformally recurrent space. (English) Zbl 0789.53012
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 617-628 (1992).
An \((n+1)\)-dimensional \((n\geq 4)\) Riemannian manifold \((\overline{M}, \overline{g})\) is called a conformally recurrent manifold if its Weyl conformal curvature tensor \(\overline{C}\) takes the form \(\overline{\nabla} \overline{C} = \overline{a} \otimes \overline{C}\) for some 1-form \(\overline{a}\), called the recurrence 1-form. A hypersurface \((M,g)\) of \(\overline{M}\) is called a quasi-umbilical hypersurface if the second fundamental form \(h\) of \(M\) in \(\overline{M}\) satisfies \(h = \alpha g + \beta v \otimes v\), where \(\alpha\) and \(\beta\) are certain functions and \(v\) is a 1-form on \(M\). When \(v\) is torse-forming, the author obtains conditions for \(M\) to be conformally recurrent with the recurrent 1-form being the pull-back of the recurrent 1-form of \(\overline{M}\). He also obtains conditions for the hypersurface to be conformally flat.
For the entire collection see [Zbl 0764.00002].
53B25 Local submanifolds
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