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.


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