Weakly η -Einstein Contact Manifolds

In this paper, we introduce the notion of weakly η -Einstein structure. Then we prove that a 3-dimensional η -Einstein almost contact metric manifold is weakly η -Einstein. Moreover, the generalized Sasakian space forms are weakly η -Einstein. Furthermore, we obtain the characteristic equation for a non-Sasakian contact (k,μ ) -space to be weakly η -Einstein, which provides many interesting examples. In particular, we determine the base manifold whose unit tangent sphere bundle T_1M(c) is weakly η -Einstein.