Geometry of space of Riemannian manifolds

设A为具有有界曲率和Hausdorff维数的Alexnadrov空间的空间，并具有Hausdorff距离，I为A上的上连续函数空间。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。由于所有网的构形空间都与空间的乘积相同，所以A及其网中的空间对的集合N可以解释为A上的纤维空间。我们考虑一个地图，分配矩阵的相互距离的两个点为每个网。这样我们就可以把N表示为某个Banach空间的子空间。然后我们引入了从N到某个欧氏空间的其他映射，这些映射获取了上述距离矩阵的局部信息。特别地，我们定义了类似于黎曼流形上函数的拉普拉斯算子的离散拉普拉斯算子。引入了一种新的统计方法，对网的位形空间上的离散拉普拉斯算子求平均。这样我们证明了离散Laplacian的特征值和特征向量收敛于极限而与网的选择无关，并证明了这在某种意义上与Laplacian在K-Machigashira-shioya意义下是一致的.其次，我们通过比较不同空间和网的两个离散Laplacian，定义了A上的新结构，因为它们是矩阵空间的同一成员.由于在信息几何中两个分布的相对熵决定了它们的Reimannian度量，我们首先从Laplacian出发引入平稳马氏链，然后将相对熵应用于它们，最后构造了它们的连续极限，它是Hausdorff距离的推广.

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.