Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 1, pp. 3-29   https://doi.org/10.15407/mag17.01.003     ( to contents , go back )

### Natural Ricci Solitons on Tangent and Unit Tangent Bundles

Laboratory of Mathematical Sciences and Applications (LASMA), Department of Mathematics, Faculty of sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdallah, B.P. 1796 Atlas, Fez, Morocco
E-mail: mtk abbassi@yahoo.fr

Noura Amri

Laboratory of Mathematical Sciences and Applications, Faculty of sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdallah, B.P. 1796 Atlas, Fez, Morocco
E-mail: amri.noura1992@gmail.com

Received November 27, 2019, revised January 20, 2020.

Abstract

Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on the tangent bundle of a flat Riemannian manifold equipped with these $g$-natural metrics. When unit tangent bundles over a constant curvature Riemannian manifold are endowed with pseudo-Riemannian Kaluza--Klein type metric, we give a classification of Ricci soliton structures whose potential vector fields are fiber-preserving, inferring the existence of some of them which are non Einstein.

Mathematics Subject Classification 2010: 53C25, 53D25
Key words: tangent bundle, unit tangent (sphere) bundle, g-natural metrics, metrics of Kaluza-Klein type, Ricci solitons