# Immersions into Statistical Manifolds

## TL;DR

This paper explores the geometric conditions for immersions into statistical manifolds, focusing on duality, realizability, and conformal-projective flatness in higher-dimensional contexts.

## Contribution

It provides necessary and sufficient conditions for dual structures, realizability of statistical manifolds, and introduces centro-affine immersions in this geometric framework.

## Key findings

- Dual statistical manifold structures characterized by specific conditions.
- Conditions established for realizing n-dimensional manifolds in (n+1)-dimensional spaces.
- Statistical manifolds immersed in dually flat spaces are conformally-projectively flat.

## Abstract

This paper studies the geometry of immersions into statistical manifolds. A necessary and sufficient condition is obtained for statistical manifold structures to be dual to each other for a non-degenerate equiaffine immersion. Then we obtain conditions for realizing an n-dimensional statistical manifold in an (n+1)-dimensional statistical manifold and its converse. Centro-affine immersion of codimension two into a dually flat statistical manifold is defined. Also we have shown that statistical manifold realized in a dually flat statistical manifold of codimension two is conformally-projectively flat.