When are the tangent sphere bundles of a Riemannian manifold eta-Einstein?

Abstract

We study the geometry of a tangent sphere bundle of a Riemannian manifold (M, g). Let M be an n-dimensional Riemannian manifold and T(r)M be the tangent bundle of M of constant radius r. The main theorem is that T(r)M equipped with the standard contact metric structure is eta-Einstein if and only if M is a space of constant sectional curvature 1/(r)2 or n-2/(r)2.