Remarks on $η$-Einstein unit tangent bundles

Abstract

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $η$-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 $η$-Einstein manifold if and only if base manifold is the space of constant sectional curvature 1 or 2.

15 pages