论文标题
仿射平坐标中EM波前的局部正常形式
Local normal forms of em-wavefronts in affine flat coordinates
论文作者
论文摘要
在我们以前的作品中，我们将双重平坦或Hessian歧管的概念推广到了Quasi-Hessian歧管。它承认Hessian指标是退化的，但具有特定的对称立方张量（广义Amari-Centsov张量）。确实，它自然是信息几何和相关字段中的单数模型。准赫西安的歧管在本地伴随着可能的多价势及其双重，其图被称为$ e $ wavefront和$ m $ - $ -Wavefront，以及连贯的切线切线捆绑包，并带有平坦的连接。在本文中，使用这些连接和指标，我们提供了无坐标的标准，用于检测$ e/m $ $ -WaveFronts的本地差异类型，然后通过使用Malgrange的Affine Flat Coordinate的$ E/M $ -WaveFronts来得出这些（双重）潜在功能的本地正常形式。这是由Ekeland关于非凸优化的早期作品以及Saji-umehara-Yamada在波前的Riemannian几何形状上的作品进行的。最后，我们揭示了几何标准与信息几何量的统计流形的关系。
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.
