#
A Foliation by Deformed Probability Simplexes for Transition of α-Parameters^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Dualistic Structures and Divergences on the Probability Simplex

## 3. Deformed Probability Simplexes and Escort Distributions

## 4. Divergences Generated by Affine Immersions as Level Surfaces

**Theorem**

**1**

**.**Let M be a simply connected n-dimensional level surface of φ on an $(n+1)$-dimensional Hessian domain $(\mathsf{\Omega},\tilde{D},\tilde{g}=\tilde{D}d\phi )$ with a Riemannian metric $\tilde{g}$ and suppose that $n\ge 2$. If we consider $(\mathsf{\Omega},\tilde{D},\tilde{g})$ a flat statistical manifold, $(M,D,g)$ is a 1-conformally flat statistical submanifold of $(\mathsf{\Omega},\tilde{D},\tilde{g})$, where D and g denote the connection and the Riemannian metric on M induced by $\tilde{D}$ and $\tilde{g}$, respectively.

**Definition**

**1**

**.**Let $(N,\nabla ,h)$ be a 1-conformally flat statistical manifold realized by a non-degenerate affine immersion $(v,\xi )$ into ${\mathbf{A}}^{n+1}$, and w the conormal immersion for v. Then, the divergence ${\rho}_{\mathrm{c}onf}$ of $(N,\nabla ,h)$ is defined by

**Theorem**

**2**

**.**For a 1-conformally flat statistical submanifold $(M,D,g)$ of $(\mathsf{\Omega},\tilde{D},\tilde{g})$, two divergences ${\rho}_{\mathrm{c}onf}$ and ${\rho}_{\mathrm{s}ub}$ coincide.

## 5. Extended Divergence on a Foliation by Deformed Probability Simplexes

**Proposition**

**1.**

**Proof.**

**Definition**

**2.**

**Proposition**

**2.**

**Proof.**

## 6. Decomposition of an Extended Divergence

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Uohashi, K. A Foliation by Deformed Probability Simplexes for Transition of *α*-Parameters. *Phys. Sci. Forum* **2022**, *5*, 53.
https://doi.org/10.3390/psf2022005053

**AMA Style**

Uohashi K. A Foliation by Deformed Probability Simplexes for Transition of *α*-Parameters. *Physical Sciences Forum*. 2022; 5(1):53.
https://doi.org/10.3390/psf2022005053

**Chicago/Turabian Style**

Uohashi, Keiko. 2022. "A Foliation by Deformed Probability Simplexes for Transition of *α*-Parameters" *Physical Sciences Forum* 5, no. 1: 53.
https://doi.org/10.3390/psf2022005053