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Zbl 1062.53035
Chaki, M.C.
On generalized quasi Einstein manifolds.
(English)
[J] Publ. Math. 58, No. 4, 683-691 (2001). ISSN 0033-3883

The author calls a nonflat Riemannian manifold $(M^n,g)$ (always $n>3$) generalized quasi Einstein if $S(X,Y)=ag(X,Y)+bA(X)A(Y)+c[A(X)B(Y)+A(Y)B(X)]$ $\forall X,Y\in \frak X (M)$, where $S$ is the Ricci tensor, $a$, $b\ne 0$, $c$ are scalars, and $A,B\in \frak X^*(M)$ are 1-forms such that their counterparts $U,V\in \frak X(M)$ are perpendicular unit vector fields. If $c=0$, then we obtain a quasi Einstein manifold $((QE)_n)$ introduced by {\it M. C. Chaki} and {\it 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 {\it B. Chen} and {\it K. Yano} [Tensor, New Ser. 26, 318--322 (1972; Zbl 0257.53027)] are now generalized to Riemannian manifolds of generalized quasi constant curvature $G(QC)_n$. \par 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.
[Lajos Tamássy (Debrecen)]
MSC 2000:
*53C25 Special Riemannian manifolds

Keywords: generalized quasi Einstein manifold;

Citations: Zbl 0257.53027; Zbl 0968.53030

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