  We study the geometric properties of the base manifold for the unit tangent
bundle satisfying the $\eta$-Einstein condition with the standard contact
metric structure. One of the main theorems is that the unit tangent bundle of
4-dimensional Einstein manifold, equipped with the canonical contact metric
structure, is $\eta$-Einstein manifold if and only if base manifold is the
space of constant sectional curvature 1 or 2.
