When are the tangent sphere bundles of a Riemannian manifold η-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 TrM be the tangent bundle of M of constant radius r. The main theorem is that TrM equipped with the standard contact metric structure is η-Einstein if and only if M is a space of constant sectional curvature \({\frac{1}{r^2}}\) or \({\frac{n-2}{r^2}}\).