Annals of Global Analysis and Geometry

, Volume 44, Issue 2, pp 91–103

Geodesic Ricci solitons on unit tangent sphere bundles


DOI: 10.1007/s10455-012-9357-6

Cite this article as:
Perrone, D. Ann Glob Anal Geom (2013) 44: 91. doi:10.1007/s10455-012-9357-6


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.


Tangent sphere bundlesRiemannian \(g\) natural metricsSasaki–Einstein metricsGeodesic flowInfinitesimal harmonic transformationRicci solitons

Mathematics Subject Classification (2000)


Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Fisica “E. De Giorgi”Universitá del Salento LecceItaly