Equiaffine immersion, projective flatness and quasi-Codazzi structure

TL;DR

This paper extends statistical manifold theory to affine differential geometry, linking equiaffine hypersurfaces with quasi-Codazzi structures and projective flatness, generalizing classical theorems.

Contribution

It introduces a new quasi-Codazzi framework within para-Hermitian geometry, characterizing equiaffine hypersurfaces via projectively flat dual connections.

Findings

Equiaffine hypersurfaces correspond to quasi-Codazzi structures with projectively flat dual connections.

The theory generalizes classical results by Weyl, Radon, Nomizu, and Kurose.

The framework applies to hypersurfaces with possibly degenerate tensors.

Abstract

In the present paper, we study an extended theory of statistical manifolds in application to affine differential geometry. Any smooth hypersurface $M \subset \mathbb{R}^{n+1}$ with a transverse vector field $\xi$ naturally admits a symmetric $(0, 2)$-tensor $h$ and a torsion-free connection $\nabla$ on $M$ so that $\nabla h$ is totally symmetric. Here $h$ may be degenerate (i.e., not a pseudo-Riemannian metric) in general. As a generalization of classical theorem due to Weyl, Radon, Nomizu, Kurose and others, we show, roughly saying, that $M$ with $\xi$ is equiaffine if and only if $(h, \nabla)$ defines a quasi-Codazzi structure, previously introduced by the author, and it admits a projectively flat dual connection with symmetric Ricci contraction. This is a direct consequence from our quasi-Codazzi theory, which is built in a more general context as a submanifold theory in para-Hermitian geometry.