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

In this study, we investigate the tangent bundle TM 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 TM 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 TM to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to TM, 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 TM 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.