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Codazzi方程，则据称是Codazzi方程。几何和仿射差异几何形状，并在信息几何形状中起重要，重要和中心作用。在这个项目中，我们从差异几何和信息几何观点上进行了针对Hessian（Codazzi）结构的基本研究，并获得了以下结果。1我们将不变的现象与Lie代数的某些仿射表示存在相关联。 Using such affine representation we provided :(1) A homogeneous space G/K admits an invariant projectively flat affine connection if and only if G/K has an equal 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 admissions an invariant Codazzi structure of constant curvature c=0 if and only if it has an equal immersion of codimension 1 into a certain homogenous Hessian manifolds.2 For a linear mapping ρ of a domain Ω into the space of positively defined symmetric matrices we constructed an exponential family of probability distributions parametrically by the elements in Riy D1nie D1×Ω, and studiod a Hessian structure on Riy D1nie D1×ω由指数族给出。使用ρ我们在紧凑的双曲线歧管上的矢量束上引入了黑森结构，并证明了某些消失的定理。