Annals of Global Analysis and Geometry

, Volume 44, Issue 2, pp 91–103 | Cite as

Geodesic Ricci solitons on unit tangent sphere bundles

Article

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.

Keywords

Tangent sphere bundles Riemannian \(g\) natural metrics Sasaki–Einstein metrics Geodesic flow Infinitesimal harmonic transformation Ricci solitons 

Mathematics Subject Classification (2000)

53C25 53D25 

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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

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