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 ${}^{C}g$. First, we define a Ricci quarter-symmetric metric connection $\overline{\mathrm{\nabla }}$ on the tangent bundle $TM$ equipped with the complete lift metric ${}^{C}g$. Second, we compute all forms of the curvature tensors of $\overline{\mathrm{\nabla }}$ and study their properties. We also define the mean connection of $\overline{\mathrm{\nabla }}$. 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 $\overline{\mathrm{\nabla }}$ are presented. Finally, we search conditions for the tangent bundle $TM$ to be locally conformally flat with respect to $\overline{\mathrm{\nabla }}$.