The author calls a nonflat Riemannian manifold (always ) generalized quasi Einstein if , where is the Ricci tensor, , , are scalars, and are 1-forms such that their counterparts are perpendicular unit vector fields. If , then we obtain a quasi Einstein manifold introduced by M. C. Chaki and K. Maity [Publ. Math. 57, 297–306 (2000; Zbl 0968.53030)]. If moreover , then we obtain an Einstein manifold. Also the Riemannian manifolds of quasi constant curvature 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 .
It is shown that: 1) and are the Ricci curvatures in the directions of , resp. ; 2) every is a ; 3) every conformally flat is a ; 4) every is . The sectional curvatures of a conformally flat are also studied.