Geometry of space of Riemannian manifolds

黎曼流形空间几何

基本信息

批准号：
11640075
负责人：
OTSU Yukio
金额：
$ 2.3万
依托单位：
Kyushu University (2001-2002)Osaka University (1999-2000)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

项目摘要

Let us denote by A the space of Alexnadrov spaces of bounded curvature below and Hausdorff dimension above equipped with Hausdorff distance and by I the space of upper-semicontinuous functions on A. We call I the space of invariants. An ordered finite set of points of metric space is called a net, which is a discretization of the metric space. Since the configuration space of all nets is identified with the product of the space, the set N of pairs of spaces in A and its nets can be interpreted as a fiber space over A. We consider a map that assign the matrix of mutual distances of two points for each net. In this way we can represent N as a subspace of some Banach space. Then we introduce other maps form N to some Euclidian space that take local information of the above distance matrix. Especially we defined discrete Laplacian similar to the Laplacian of functions of Riemannian manifold. We introduced new statistical method to take average of discrete Laplacian on configuration space of nets. In this way we have showd that the eigenvalues and eigenvectors of discrete Laplacian converge to the limit independent of the choice of nets ; we also proved that coincides with the Laplacian in the sense of Kuwae-Machigashira-shioya in some sense.Next we defined new structure on A by comparing two discrete Laplacian of different spaces and nets because they are same member of matrix space. Since in information geometry the relative entropy of two distributions determines Reimannian metric, we first introduced stationary Markov chain form the Laplacian, then we apply the relative entropy for them; finally we construct continuum limit of them, which is a generalization of Hausdorff distance.

让我们用 A 表示下面有界曲率的 Alexnadrov 空间和上面配备 Hausdorff 距离的 Hausdorff 维数的空间，用 I 表示 A 上的上半连续函数的空间。我们称 I 为不变量空间。度量空间的有序有限点集称为网，它是度量空间的离散化。由于所有网络的配置空间都由空间的乘积来确定，因此 A 及其网络中的空间对的集合 N 可以解释为 A 上的光纤空间。我们考虑一个映射，该映射分配相互距离的矩阵每网两分。这样我们就可以将 N 表示为某个 Banach 空间的子空间。然后我们引入从 N 到某个欧几里得空间的其他映射，这些映射采用上述距离矩阵的局部信息。特别是我们定义了类似于黎曼流形函数拉普拉斯的离散拉普拉斯算子。我们引入了新的统计方法来在网络的配置空间上取离散拉普拉斯算子的平均值。通过这种方式，我们证明了离散拉普拉斯算子的特征值和特征向量收敛到极限，与网络的选择无关；我们还证明了在某种意义上与 Kuwae-Machigashira-shioya 意义上的拉普拉斯算子一致。接下来，我们通过比较不同空间和网络的两个离散拉普拉斯算子来定义 A 上的新结构，因为它们是矩阵空间的相同成员。由于在信息几何中，两个分布的相对熵决定了雷曼度量，因此我们首先从拉普拉斯算子中引入平稳马尔可夫链，然后对它们应用相对熵；最后我们构造了它们的连续极限，这是豪斯多夫距离的推广。