New Results on Einstein and $\eta$-Einstein tangent sphere bundles

Abstract

We complete the results from [14] concerning tangent sphere bundles of constant radius $r$, previously valid for nonflat base manifolds of dimension $n>2$, by taking as base manifold an arbitrary $2$-dimensional space form$(M,g)$. We derive expressions for the metric ${}^rG$ induced on $T_rM$ by a natural diagonal metric from $TM$, such that $(T_rM,{}^rG)$ is an Einstein manifold, i.e. its Ricci tensor field is $Ric=\rho\cdot{}^rG$. In particular, if the metric is induced by the Sasaki metric, $T_rM$ is an Einstein manifold if and only if $(M,g)$ has positive constant sectional curvature $c$, $r=\frac{\sqrt{c}}{c}$, $\rho=\frac{c}{2}$, or $(M,g)$ is flat, and then $T_rM$ is Ricci-flat. Moreover, we show that for a flat $n$-dimensional Riemannian manifold $(M,g)$, $T_rM$ is Einstein with respect to the metric induced by the natural diagonal metric from $TM$ if and only if $n=2$, $(T_rM,{}^rG)$ is Ricci flat, and a certain function coefficient of the metric vanishes. Finally, we consider the Sasakian manifold $(T_rM,\varphi,\xi,\eta,G)$ from \cite{DrutaOpr} and provide a simpler characterization of the $\eta$--Einstein Sasakian tangent sphere bundles in terms of rational expressions involving the coefficients of the metric, the dimension of the base manifold, and the radius $r$.