Local normal forms of em-wavefronts in affine flat coordinates

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