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Zbl 1148.53012
Głogowska, Małgorzata
On quasi-Einstein Cartan type hypersurfaces. (English)
[J] J. Geom. Phys. 58, No. 5, 599-614 (2008). ISSN 0393-0440

The paper under review studies curvature properties of quasi-Einstein Cartan type hypersurfaces in semi-Riemannian manifolds. The main result shows that the Rieman curvature tensor $R$, Weyl conformal curvature tensor $C$, and Ricci curvature tensor $S$ of a quasi-Einstein Cartan type hypersurface in a semi-Riemannian space form with scalar curvature $\tilde{\kappa}$ satisfy one of the following conditions: (a) $R\cdot R=\frac{\tilde{\kappa}}{n(n+1)}Q(g,R),\;\; C\cdot C=\frac{n-3}{2(n-2)}(\frac{\tilde{\kappa}}{(n+1)}-\frac{\tilde{\kappa}}{(n-1)})Q(g,C)$, or (b) $R\cdot S=\frac{\tilde{\kappa}}{n(n+1)}Q(g,S),\;\;R\cdot R-\frac{\tilde{\kappa}}{n(n+1)}Q(g,R)\ne 0$, and $C\cdot C$ and $Q(g,C)$ are linearly independent, or (c) $R\cdot S-\frac{\tilde{\kappa}}{n(n+1)}Q(g,S)\ne 0$ and the hypersurface is $2$-quasi-umbilical on some neighborhood with $C\cdot C=L_CQ(g,C)$ holds on this set.
[Ye-Lin Ou (Commerce)]

MSC 2000:: *53B20 Local Riemannian geometry
53B25 Local submanifolds
53B30 Lorentz metrics, indefinite metrics

Keywords: pseudosymmetry type manifold; Cartan type hypersurface; quasi-Einstein type hypersurface

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