Statistical structures on tangent bundles and tangent Lie groups

Abstract

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.

Quasi-Statistical Schouten–van Kampen Connections on the Tangent Bundle

“…The results obtained in this work lead to new examples of (quasi-)statistical structures on the tangent bundle of a Riemann manifold. Unlike the majority of previous studies (see, e.g., [4,13,19,22,24]), which produce new examples of statistical structures on the tangent bundle by lifting a given statistical structure on the base space, the present article does not assume the a priori existence of a statistical structure on the base manifold. The new structures are, thus, uncorrelated with the ones from the base, therefore constituting a more convenient geometric setting to investigate the statistical behavior in depth.…”

“…Statistical structures on the tangent bundle of differentiable manifolds were treated in recent papers, such as [4,13,19,22,24].…”

Statistical structures and Killing vector fields on tangent bundles with respect to two different metrics

“…Although curvature related properties of tangent bundles are widely studied, investigating statistical structures on tangent bundles is a relatively new topic. These structures were examined with respect to various Riemannian metrics such as the Sasaki metric [5], the Cheeger-Gromoll metric and a g−natural metric which consists of three classic lifts of the metric g [12], the twisted Sasaki metric and the gradient Sasaki metric [11].…”

Analyzing Curvature Properties and Geometric Solitons of the Twisted Sasaki Metric on the Tangent Bundle over a Statistical Manifold

“…Matsuzoe provided an overview of the geometry of statistical manifolds and discussed the connections between information geometry and affine differential geometry [41]. In [42], Peyghan, Seifipour, and Gezer investigated statistical structures on the tangent bundle TM equipped with two Riemannian g-natural metrics and lift connections.…”