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

Abstract

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{\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{\nabla } $ and study their properties. We also define the mean connection of $ \overline{\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{\nabla } $ are presented. Finally, we search conditions for the tangent bundle $ TM $ to be locally conformally flat with respect to $ \overline{\nabla } $.

Keywords

• complete lift metric
• ricci quarter-symmetric metric connection
• ricci soliton
• tangent bundle
• vector field