  A rigidity result for a class of compact generalized quasi-Einstein manifolds
with constant scalar curvature is obtained. Moreover, under some geometric
assumptions, the rigidity for the noncompact case is also proved. Considering
non constant scalar curvature, we characterize the generalized quasi-Einstein
manifolds which are conformal to the Euclidean space and we show that there
exist two classes of complete manifolds, which are obtained by considering
potential functions and conformal factors either to be radial or invariant
under the action of an (n-1)-dimensional translation group. Explicit examples
are given.
