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Zbl 1026.53010
Deszcz, Ryszard; Głogowska, Małgorzata
Examples of nonsemisymmetric Ricci-semisymmetric hypersurfaces.
(English)
[J] Colloq. Math. 94, No.1, 87-101 (2002). ISSN 0010-1354; ISSN 1730-6302/e

Suppose $\left( M^n,g\right)$ is a semi-Riemannian manifold of dimension $n\geq 3$. $\left( M^n,g\right)$ is said to be semisymmetric if $R\cdot R=0$, and Ricci-semisymmetric if $R\cdot S=0$, where $R$ is the Riemann-Christoffel curvature tensor and $S$ is the Ricci tensor on the manifold. In general the first condition implies the second, but not conversely. In this paper the authors construct a class of Ricci-semisymmetric warped products which are not semisymmetric. Examples of Ricci-semisymmetric, non-semisymmetric hypersurfaces in semi-Euclidean space $E_s^{n+1}$ for $n\geq 5$ are given as well.
[Lew Friedland (Geneseo)]
MSC 2000:
*53B30 Lorentz metrics, indefinite metrics
53B25 Local submanifolds
53C40 Submanifolds (differential geometry)
53C35 Symmetric spaces (differential geometry)
53B50 Appl. of local differential geometry to physics
53B20 Local Riemannian geometry

Keywords: Ricci-semisymmetric manifold; quasi-Einstein manifold; warped product; hypersurface; Cartan hypersurface; P. J. Ryan problem

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