The tangent sphere bundle of a surface in terms of the Ricci tensor (Q2782741)

Consider the unit tangent bundle \(T_1M\) of a \(2\)-dimensional Riemannian manifold \(M\) equipped with the Riemannian metric \(g\) which is induced from the Sasaki metric on the tangent bundle of \(M\). The authors prove the following results. The Ricci tensor of \((T_1M,g)\) is a Killing tensor if and only if \(M\) has constant Gaussian curvature. The Ricci tensor of \((T_1M,g)\) is a Codazzi tensor if and only if \(M\) is flat or if \(M\) has constant Gaussian curvature one. \((T_1M,g)\) is conformally flat if and only if \(M\) is flat or has constant Gaussian curvature one.