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

Let $ (M, g) $ be an n-dimensional (pseudo-)Riemannian manifold and $ TM $ be its tangent bundle $ TM $ equipped with the complete lift metric Cg. First, we define a Ricci quarter-symmetric metric connection ∇ ̄ on the tangent bundle $ TM $ equipped with the complete lift metric Cg. 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 $ TM $ to become a Ricci soliton and a gradient Ricci soliton concerning ∇ ̄ are presented. Finally, we search conditions for the tangent bundle $ TM $ to be locally conformally flat with respect to ∇ ̄.