Geometry of tangent bundles of statistical manifolds equiped with Cheeger-Gromoll type metrics

TL;DR

This paper explores the geometric properties of tangent bundles of statistical manifolds equipped with a family of Cheeger-Gromoll type metrics, including curvature, geodesics, and conditions for constant sectional curvature.

Contribution

It provides explicit formulas for Levi-Civita connections and curvature, analyzes geodesic behavior, and characterizes when tangent bundles have constant sectional curvature.

Findings

Derived the Levi-Civita connection and curvature expressions for the tangent bundle.

Identified conditions for fibers to be totally geodesic and for geodesic flow to be incompressible.

Established criteria for the tangent bundle to have constant sectional curvature.

Abstract

In this paper, we investigate the geometry of the tangent bundle $TM$ of a statistical manifold $(M,g,\nabla)$ endowed with a two-parameter family of generalized Cheeger--Gromoll metrics $g_{p,q}$. We compute the associated the Levi--Civita connection $\nabla^{p,q}$ and express its curvature in terms of the Riemannian curvature and the skewness tensor $K$ of the base statistical manifold. We further analyze the behavior of geodesics, identify conditions under which the fibers of $TM$ are totally geodesic, and determine when the geodesic flow associated with $g_{p,q}$ is incompressible. Moreover, we establish necessary and sufficient conditions for the tangent bundle to admit constant sectional curvature. Several examples are provided to illustrate the theory, including statistically deformed Euclidean spaces and information geometric models such as the manifold of normal distributions. The sectional curvature of $(TM, g_{p,q})$ is computed for horizontal, vertical, and mixed directions, leading to a concise expression for the corresponding scalar curvature.