Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections

In this study, we investigate the tangent bundle T M of an n -dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection ∇ ˜ . Our primary goal is to establish the necessary and sufficient conditions for T M to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for T M to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to T M , alongside specific constraints on the conformal factor λ and the geometric properties of M . For gradient conformal Yamabe solitons, the conditions involve λ and the Hessian of the potential function. Similarly, for T M to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of λ , and the curvature properties of M . For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M . These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry.