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.
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References
Abbassi, K.M.T.: \(g\)-natural metrics: new horizons in the geometry of tangent bundles of Riemannian manifolds. Note Mat 28(1), 6–35 (2008)
Abbassi, K.M.T., Kowalski, O.: On Einstein Riemannian \(g\)-natural metrics on unit tangent sphere bundles. Ann. Glob. Anal. Geom. 38, 11–20 (2010)
Abbassi, K.M.T., Sarih, M.: On some hereditary properties of Riemannian \(g\)-natural metrics on tangent bundles of Riemannian manifolds. Differ. Geom. Appl. 22(1), 19–47 (2005)
Abbassi, K.M.T., Calvaruso, G.: \(g\)-natural contact metrics on unit tangent sphere bundles. Monatsh. Math. 151, 189–209 (2006)
Abbassi, K.M.T., Calvaruso, G., Perrone, D.: Harmonicity of unit vector fields with respect to Riemannian natural metrics. Differ. Geom. Appl. 27, 157–169 (2009)
Abbassi, K.M.T., Calvaruso, G., Perrone, D.: Harmonic maps defined by the geodesic flow. Houston J. Math. 36(1), 69–90 (2010)
Abbassi, K.M.T., Calvaruso, G., Perrone, D.: Harmonic sections of tangent bundles equipped with Riemannian \(g\)-natural metrics. Quart. J. Math. 62, 259–288 (2011)
Benyounes, M., Loubeau, E., Todjihounde, L.: Harmonic maps and Kaluza–Klein metrics on Spheres. Rocky Mountain. J. Math. 42, 791–821 (2012)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds Progress in Mathematics, vol. 203, 2nd edn. Birkhäuser, Boston (2010)
Boyer, C., Galicki, K.: Sasakian Geometry: Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)
Cao, H.D.: Geometry of Ricci solitons. Chinese Ann. Math. Ser. B 27B, 121–142 (2006)
Calvaruso, G., Perrone, D.: Homogeneous and H-contact unit tangent sphere bundles. J. Aust. Math. Soc. 88, 323–337 (2010)
Calvio-Louzao, E., Seoane-Bascoy, J., Vzquez-Abal, M.E., Vzquez-Lorenzo, R.: One-harmonic invariant vector fields on three-dimensional Lie groups. J. Geom. Phys. 62, 1532–1547 (2012)
Cho, J.T.: Notes on contact Ricci solitons. Proc. Edinb. Math. Soc. 54, 47–53 (2011)
Chow, B., Knopf, D.: The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, vol. 110. American Mathematical Society, Providence (2004)
Crasmareanu, M.: Liouville and Ricci solitons. C. R. Acad. Sc. Paris Ser. I. 347, 1305–1308 (2009)
Dragomir, S., Perrone, D.: Harmonic Vector Fields: Variational Principles and Differential Geometry. Elsevier, Science Ltd., (2011)
Kolář, I., Michor, P.W., Slovák, J.: Natural operations in differential geometry. Springer, Berlin (1993)
Hermann, R.: Yang–Mills, Kaluza–Klein, and the Einstein Program: Interdisciplinary Math., vol. XIX. Math Sci Press, Brookline (1978)
Kowalski, O., Sekizawa, M.: Natural Transformations of Riemannian Metrics on Manifolds to Metrics on Tangent Bundles: A Classification. Bull. Tokyo Gakugei Univ. IV 40, 1–29 (1988)
Nouhaud, O.: Transformations infinitesimales harmoniques. C. R. Acad. Sci. Paris Ser. A. 274, 573–576 (1972)
Perrone, D.: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Differ. Geom. Appl. 20, 367–378 (2004)
Perrone, D.: Minimality, harmonicity and CR geometry for Reeb vector fields. Int. J. Math. 21(9), 1189–1218 (2010)
Stepanov, S.E., Shandra, I.G.: Geometry of infinitesimal harmonic transformations. Ann. Global Anal. Geom. 24, 291–299 (2003)
Stepanov, S.E., Shelepova, V.N.: A note on Ricci solitons. Math. Notes 86(3), 447–450 (2009)
Szab\(\acute{o}\), Z.I.: A short topological proof for the symmetry of 2 point homogeneous spaces. Invent. Math. 106, 61–64 (1991)
Wang, H.C.: Two point homogeneous spaces. Ann. of Math. 55, 177–191 (1952)
Wood, C.M.: An existence theorem for harmonic sections. Manuscripta Math. 68, 69–75 (1990)
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This study was supported by Universitá del Salento and M.I.U.R. (within P.R.I.N).
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Perrone, D. Geodesic Ricci solitons on unit tangent sphere bundles. Ann Glob Anal Geom 44, 91–103 (2013). https://doi.org/10.1007/s10455-012-9357-6
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DOI: https://doi.org/10.1007/s10455-012-9357-6
Keywords
- Tangent sphere bundles
- Riemannian \(g\) natural metrics
- Sasaki–Einstein metrics
- Geodesic flow
- Infinitesimal harmonic transformation
- Ricci solitons