Differential Geometry and Information Geometry II

微分几何与信息几何II

基本信息

批准号：
10440022
负责人：
SHIMA Hirohiko
金额：
$ 4.22万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

A pair (D, g) of an affine connection D and a Riemann metric g is said to be a Codazzi structure if g satisfies Codazzi equattion with respect to D. For a Codazzi structure (D, g) , D is flat if and only if (D, g) is a Hessian structure, that is, g is locally expressed by a Hessian with respect to affine coordinate systems for D. Hessian (Codazzi) structures are deeply connected with Kahler geometry and affine differential geometry, and play important, essential and central roles in information geometry. In this project we engaged in fundamental researches for Hessian (Codazzi) structures from both differential geometric and information geometric viewpoints, and obtained the following results.1 We relate the existence of invariant projectively flat affine connections to that of certain affine representation of Lie algebras. Using such affine representation we proved :(1) A homogeneous space G/K admits an invariant projectively flat affine connection if and only if G/K has an equivariant centro-affine hypersurface immersion.(2) There is a bijective correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.(3) A homogeneous space admits an invariant Codazzi structure of constant curvature c=0 if and only if it has an equvariant immersion of codimension 1 into a certain homogenous Hessian manifolds.2 For a linear mapping ρ of a domain Ω into the space of positive definite symmetric matrices we conatructed an exponential family of probability distributions parametrized by the elements in RィイD1nィエD1×Ω, and studied a Hessian structure on RィイD1nィエD1×Ω given by the exponential family. Using ρ we introduced a Hessian structure on a vector bundle over a compact hyperbolic affina manifold and proved a certain vanishing theorem.

如果 g 满足关于 D 的 Codazzi 方程，则仿射连接 D 和黎曼度量 g 的一对 (D, g) 称为 Codazzi 结构。对于 Codazzi 结构 (D, g) ，D 是平坦的当且仅如果 (D, g) 是 Hessian 结构，即 g 相对于 D 的仿射坐标系局部用 Hessian 表示。 Hessian (Codazzi)结构与卡勒几何和仿射微分几何有着密切的联系，在信息几何中发挥着重要、本质和核心的作用。本课题从微分几何和信息几何的角度对Hessian(Codazzi)结构进行了基础研究。并得到以下结果。 1 我们将不变射影平坦仿射联系的存在性与李代数的某些仿射表示联系起来，使用这种仿射表示我们证明了：（1）A。齐次空间 G/K 承认不变射影平坦仿射连接当且仅当 G/K 具有等变中心仿射超曲面浸没。 (2) 具有不变射影平坦仿射连接的半单对称空间与中心空间之间存在双射对应-简单乔丹代数。(3) 齐次空间允许常曲率 c=0 的不变 Codazzi 结构当且仅当它有余维 1 等变浸入某个齐次 Hessian 流形中。2 对于域 Ω 到正定对称矩阵空间的线性映射，我们构建了由 R ×Ω 中的元素参数化的概率分布指数族，并研究了由指数族给出的 RiiD1nieD1×Ω 上的 Hessian 结构使用 ρ 我们引入了一个紧双曲仿射流形上的向量丛上的 Hessian 结构并证明了某个消失定理。