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 $$S(X,Y)=ag(X,Y)+bA(X)A(Y)+c[A(X)B(Y)+A(Y)B(X)]$$ $$\forall X,Y\in \mathfrak X (M)$$, where $$S$$ is the Ricci tensor, $$a$$, $$b\neq 0$$, $$c$$ are scalars, and $$A,B\in \mathfrak X^*(M)$$ are 1-forms such that their counterparts $$U,V\in \mathfrak X(M)$$ 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:

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

Citations:

Zbl 0257.53027; Zbl 0968.53030