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On certain quasi-Einstein semisymmetric hypersurfaces. (English) Zbl 0955.53029
Particular semisymmetric manifolds are Ricci semi-symmetric, that is semi-Riemannian manifolds $\left(M,g\right)$ for which $R·S=0$, where $R$ and $S$ are the Riemannian curvature and the Ricci tensor, respectively. Generalizing Einstein manifolds, the quasi-Einstein manifolds satisfy $S=\frac{k}{n-1}g+fw\otimes w$, where $k$ is the scalar curvature, $w$ is a 1-form and $f$ is a function. The main result here states that any quasi-Einstein Ricci-semisymmetric hypersurface of a semi-Euclidean space satisfies: $R·C=Q\left(S,C\right)$, where $C$ is the Weyl tensor. The class of Ricci-simple (i.e., $\text{rank}\phantom{\rule{4.pt}{0ex}}S\le 1$) semisymmetric manifolds is characterized and certain quasi-Einstein semisymmetric hypersurfaces are shown to be in this class.
##### MSC:
 53C40 Global submanifolds (differential geometry) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)