Natural Ricci Solitons on tangent and unit tangent bundles

TL;DR

This paper studies Ricci solitons on tangent and unit tangent bundles with pseudo-Riemannian metrics, showing their properties relate closely to the base manifold's geometry and classifying such structures under specific conditions.

Contribution

It proves Ricci soliton structures on tangent bundles induce similar structures on the base manifold and classifies these structures for certain metrics and conditions.

Findings

Tangent bundle Ricci solitons imply base manifold flatness.

Ricci solitons on tangent bundles correspond to potential fields being lifts of conformal vector fields.

Existence of non-Einstein Ricci solitons on unit tangent bundles with specific metrics.

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 a flat Riemannian manifold. 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.