Geodesic Ricci solitons on unit tangent sphere bundles

Abstract

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton$(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta$-Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.