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 T G with the left invariant Riemannian metric introduced in the paper of Asgari and Salimi Moghaddam. We investigate Einstein-like, Ricci soliton, and Yamabe soliton structures on T G. Then we study some geometrical tensors on T G such as Cotton, Schouten, Weyl, and Bach tensors, and we also compute projective and concircular and m-projective curvatures on T G. Finally, we compute the Szabo operator and Jacobi operator on the tangent Lie group T G .