  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.
