Equiaffine immersion, projective flatness and quasi-Codazzi structure
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.