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On generalized quasi Einstein manifolds. (English) Zbl 1062.53035

The author calls a nonflat Riemannian manifold (M n ,g) (always n>3) generalized quasi Einstein if S(X,Y)=ag(X,Y)+bA(X)A(Y)+c[A(X)B(Y)+A(Y)B(X)] X,Y𝔛(M), where S is the Ricci tensor, a, b0, c are scalars, and A,B𝔛 * (M) are 1-forms such that their counterparts U,V𝔛(M) are perpendicular unit vector fields. If c=0, then we obtain a quasi Einstein manifold ((QE) n ) introduced by M. C. Chaki and K. Maity [Publ. Math. 57, 297–306 (2000; Zbl 0968.53030)]. If moreover b=0, then we obtain an Einstein manifold. Also the Riemannian manifolds of quasi constant curvature (QC) n introduced by B. Chen and K. Yano [Tensor, New Ser. 26, 318–322 (1972; Zbl 0257.53027)] are now generalized to Riemannian manifolds of generalized quasi constant curvature G(QC) n .

It is shown that: 1) a and a+b are the Ricci curvatures in the directions of V, resp. U; 2) every G(QE) 3 is a G(QC) 3 ; 3) every conformally flat G(QE) n is a G(QC) n ; 4) every G(QC) n is G(QE) n . The sectional curvatures of a conformally flat G(QE) n are also studied.


MSC:
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)