Journal Article

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

Li, Yanlin, Karakaş, Erkan, Gezer, Aydin

2023

Overview

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 } $ .