Affine and Conformal Submersions with Horizontal Distribution and Statistical Manifolds

TL;DR

This paper explores the geometric properties of affine and conformal submersions with horizontal distributions, establishing conditions under which the base manifold and tangent bundle inherit statistical manifold structures.

Contribution

It introduces the concept of conformal submersion with horizontal distribution and provides necessary and sufficient conditions for statistical manifold structures to be preserved.

Findings

Base manifold inherits statistical structure under affine submersion.

Conditions for tangent bundle to be a statistical manifold with Sasaki metric.

Characterization of geodesics under conformal submersion.

Abstract

We show that, for an affine submersion $\pi: \mathbf{M}\longrightarrow \mathbf{B}$ with horizontal distribution, $\mathbf{B}$ is a statistical manifold with the metric and connection induced from the statistical manifold $\mathbf{M}$. The concept of conformal submersion with horizontal distribution is introduced, which is a generalization of affine submersion with horizontal distribution. Then proved a necessary and sufficient condition for $(\mathbf{M}, \nabla, g_M)$ to become a statistical manifold for a conformal submersion with horizontal distribution. A necessary and sufficient condition is obtained for the curve $\pi \circ \sigma$ to be a geodesic of $\mathbf{B}$, if $\sigma$ is a geodesic of $\mathbf{M}$ for $\pi: (\mathbf{M},\nabla) \longrightarrow (\mathbf{B},\nabla^*) $ a conformal submersion with horizontal distribution. Also, we obtained a necessary and sufficient condition for the tangent bundle $T\mathbf{M}$ to become a statistical manifold with respect to the Sasaki lift metric and the complete lift connection.