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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].
MSC:
 53B25 Local submanifolds