Some properties of Riemannian geometry of the tangent bundle of Lie groups

Abstract

We consider a bi-invariant Lie group (G, g) and we equip its tangent bundle TG with the left invariantRiemannian metric introduced in the paper of Asgari and Salimi Moghaddam. We investigate Einstein-like, Ricci soliton,and Yamabe soliton structures on TG. Then we study some geometrical tensors on TG such as Cotton, Schouten,Weyl, and Bach tensors, and we also compute projective and concircular and m-projective curvatures on TG. Finally,we compute the Szabo operator and Jacobi operator on the tangent Lie group TG.