Hessian Geometry and Information Geometry

海森几何与信息几何

基本信息

批准号：
15540080
负责人：
SHIMA Hirohiko
金额：
$ 2.18万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

Let M be a flat manifold with flat connection D. A Riemannian metric g on M is said to be a Hessian metric if it is locally expressed by the Hessian with respect to the flat connection D. Hessian geometry (the geometry of Hessian manifolds) is a very close relative of Kahlerian geometry, and may be placed among, and finds connection with important pure mathematical fields such as affine differential geometry, homogeneous spaces, cohomology and others. Moreover, Hessian geometry, as well as being connected with these pure mathematical areas, also, perhaps surprisingly, finds deep connections with information geometry. The notion of flat dual connections, which plays an important role in information geometry, appears in precisely the same way for our Hessian structures. Thus Hessian geometry offers both an interesting and fruitful area of research. In this project we study Hessian geometry putting together Kahlerian geometry, affine differential geometry and information geometry, and obtained the following results.1.We constructed new Hessian metrics applying a method of information geometry. Conversely, we obtained families of probability distributions using a differential geometric method.2.We developed affine differential geometry of level surfaces of potential functions of Hessian metrics, and investigating Laplacians of gradient mappings we proved a certain problem similar to the affine Bernstein's problem proposed by S.S. Chern.3.We obtained a duality theorem and vanishing theorems for Hessian manifolds similar to that of Kahlerian geometry.4.Since a Hessian structure satisfies the Codazzi equation, the notion of Hessian structures is naturally extended to the Codazzi structures. We proved that a manifold with a constant Codazzi structure has an immersion into a certain homogeneous Hessian manifold of codimension 1.

设 M 为具有平坦连接 D 的平坦流形。如果 M 上的黎曼度量 g 用关于平坦连接 D 的 Hessian 局部表示，则称其为 Hessian 度量。Hessian 几何（Hessian 流形的几何）为它是卡勒几何的近亲，可以被放置在重要的纯数学领域中，并与仿射微分几何、齐次空间、上同调等相关。此外，海森几何除了与这些纯数学领域相关之外，还令人惊讶地发现与信息几何的深层联系。平面对偶连接的概念在信息几何中起着重要作用，对于我们的 Hessian 结构也以完全相同的方式出现。因此，黑森几何提供了一个有趣且富有成果的研究领域。本课题将卡勒几何、仿射微分几何和信息几何结合起来研究Hessian几何，得到了以下成果：1.应用信息几何的方法构造了新的Hessian度量。相反，我们使用微分几何方法获得了概率分布族。 2.我们发展了Hessian度量势函数水平面的仿射微分几何，并研究了梯度映射的拉普拉斯算子，我们证明了与仿射伯恩斯坦问题类似的某个问题。 S.S. Chern.3.我们得到了类似于卡勒几何的Hessian流形的对偶定理和消失定理。4.由于Hessian结构满足Codazzi方程，Hessian结构的概念自然地扩展到Codazzi结构。我们证明了具有常 Codazzi 结构的流形浸入到余维 1 的某个齐次 Hessian 流形中。