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On quasi-Einstein Cartan type hypersurfaces. (English) Zbl 1148.53012
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 κ ˜ satisfy one of the following conditions: (a) R·R=κ ˜ n(n+1)Q(g,R),C·C=n-3 2(n-2)(κ ˜ (n+1)-κ ˜ (n-1))Q(g,C), or (b) R·S=κ ˜ n(n+1)Q(g,S),R·R-κ ˜ n(n+1)Q(g,R)0, and C·C and Q(g,C) are linearly independent, or (c) R·S-κ ˜ n(n+1)Q(g,S)0 and the hypersurface is 2-quasi-umbilical on some neighborhood with C·C=L C Q(g,C) holds on this set.
MSC:
53B20Local Riemannian geometry
53B25Local submanifolds
53B30Lorentz metrics, indefinite metrics
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