Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection

Let (M, g) be an n-dimensional (pseudo-)Riemannian manifold and T M be its tangent bundle T M equipped with the complete lift metricC g. First, we define a Ricci quarter-symmetric metric connection ∇ on the tangent bundle T M equipped with the complete lift metricC g. Second, we compute all forms of the curvature tensors of ∇ and study their properties. We also define the mean connection of ∇. Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle T M to become a Ricci soliton and a gradient Ricci soliton concerning ∇ are presented. Finally, we search conditions for the tangent bundle T M to be locally conformally flat with respect to ∇. with respect to the induced coordinates, where ∂h =∂, ∂ ∂xhh =∂ and Γh∂yh jk are the coefficients of the connection ∇. Through these lifts and the connection ∇, we can introduce on each induced coordinate neighbourhood π−1 (U) of T M a frame field which consists of the following 2n linearly independent vector fields.