Let T M be a tangent bundle over a Riemannian manifold M with a Riemannian metric g and T G be a tangent Lie group over a Lie group with a left-invariant metric g. The purpose of the paper is two folds. Firstly, we study statistical structures on the tangent bundle T M equipped with two Riemannian g-natural metrics and lift connections. Secondly, we define a left-invariant complete lift connection on the tangent Lie group T G equipped with metric g introduced in [F. Asgari and H. R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo II. Series, 2018] and study statistical structures in this setting.
We study lift metrics and lift connections on the tangent bundle T M of a Riemannian manifold (M, g). We also investigate the statistical and Codazzi couples of T M and their consequences on the geometry of M . Finally, we prove a result on 1-Stein and Osserman structures on T M , whenever T M is equipped with a complete (respectively, horizontal) lift connection.
The purpose of this paper is to find some conditions under which the tangent bundle TM has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle TM and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle TM (the complete, horizontal, and Sasaki connections) and we study the vanishing of them.
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 .
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