The Bonnet theorem for statistical manifolds

TL;DR

This paper extends the Bonnet theorem to statistical manifolds, showing conditions under which such manifolds can be locally embedded into flat or Hessian manifolds, using Lauritzen's statistical embedding framework.

Contribution

It proves the Bonnet theorem for statistical manifolds and explores embeddings with specific codimensions based on Lauritzen's embedding method.

Findings

Statistical manifolds satisfying Gauss--Codazzi--Ricci equations are locally embeddable.

Manifolds with affine embeddings of codimension 1 or 2 are embeddable into flat statistical manifolds.

The proof utilizes Lauritzen's notion of statistical embedding.

Abstract

We prove the Bonnet theorem for statistical manifolds, which states that if a statistical manifold admits tensors satisfying the Gauss--Codazzi--Ricci equations, then it is locally embeddable to a flat statistical manifold (or a Hessian manifold). The proof is based on the notion of statistical embedding to the product of a vector space and its dual space introduced by Lauritzen. As another application of Lauritzen's embedding, we show that a statistical manifold admitting an affine embedding of codimension 1 or 2 is locally embeddable to a flat statistical manifold of the same codimension.