Classification of Vector Fields and Soliton Structures on a Tangent Bundle with a Ricci Quarter-Symmetric Metric Connection

Consider TM as the tangent bundle of a (pseudo-)Riemannian manifold M, equipped with a Ricci quarter-symmetric metric connection (Formula presented.). This research article aims to accomplish two primary objectives. Firstly, the paper undertakes the classification of specific types of vector fields, including incompressible vector fields, harmonic vector fields, concurrent vector fields, conformal vector fields, projective vector fields, and (Formula presented.)(Ric) vector fields within the framework of (Formula presented.) on (Formula presented.). Secondly, the paper establishes the necessary and sufficient conditions for the tangent bundle TM to become as a Riemannian soliton and a generalized Ricci-Yamabe soliton with regard to the connection (Formula presented.).