Local normal forms of em-wavefronts in affine flat coordinates

TL;DR

This paper develops coordinate-free criteria and local normal forms for wavefronts in quasi-Hessian manifolds, linking geometric structures with information geometry and singularity theory.

Contribution

It introduces new criteria for classifying wavefronts and derives their normal forms in affine coordinates, extending the understanding of quasi-Hessian manifolds.

Findings

Coordinate-free criteria for wavefront types

Normal forms of potential functions in affine coordinates

Connection between geometric criteria and information geometry

Abstract

In our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold; it admits the Hessian metric to be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, it naturally appears as a singular model in information geometry and related fields. A quasi-Hessian manifold is locally accompanied with a possibly multi-valued potential and its dual, whose graphs are called the $e$-wavefront and the $m$-wavefront respectively, together with coherent tangent bundles endowed with flat connections. In the present paper, using those connections and the metric, we give coordinate-free criteria for detecting local diffeomorphic types of $e/m$-wavefronts, and then derive the local normal forms of those (dual) potential functions for the $e/m$-wavefronts in affine flat coordinates by means of Malgrange's division theorem. This is motivated by an early work of Ekeland on non-convex optimization and Saji-Umehara-Yamada's work on Riemannian geometry of wavefronts. Finally, we reveal a relation of our geometric criteria with information geometric quantities of statistical manifolds.