Weakly η-Einstein Contact Manifolds

In this paper, we introduce the notion of weakly eta-Einstein structure. Then we prove that a 3-dimensional eta-Einstein almost contact metric manifold is weakly eta-Einstein. Moreover, the generalized Sasakian space forms are weakly eta-Einstein. Furthermore, we obtain the characteristic equation for a non-Sasakian contact (k, mu)-space to be weakly eta-Einstein, which provides many interesting examples. In particular, we determine the base manifold whose unit tangent sphere bundle T1M(c) is weakly eta-Einstein.