Affine and Conformal Submersions with Horizontal Distribution and Statistical Manifolds

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.