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On some properties of generalized quasi-Einstein manifolds. (English) Zbl 1168.53026

The authors consider generalized quasi-Einstein manifolds. A non flat Riemannian manifold \((M^n,g)\) is called generalized quasi-Einstein if its Ricci tensor \(S\) satisfies the condition \[ S(X,Y)=ag(X,Y)+bA(X)A(Y)+cB(X)B(Y), \]
where \(a,b,c\) are non-zero functions and \(A,B\) non-zero 1-forms. The authors find the necessary condition for a generalized quasi-Einstein manifold to be Ricci-pseudosymmetric in the sense of R.Deszcz. Moreover, they prove that a 2-quasi umbilical hypersurface in a Riemannian space form is a generalized quasi-Einstein manifold. Finally, they give two examples of such hypersurfaces.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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