Geometry of space of Riemannian manifolds

设A表示赋Hausdorff距离的下方有界曲率的Alexnadrov空间和上方的Hausdorff维的Alexnadrov空间，I表示A上的上半连续函数空间，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。由于所有网的构型空间都用该空间的乘积来表示，因此A及其网的空间对集合N可以解释为A上的纤维空间。我们考虑一个映射，它为每个网分配两点的相互距离矩阵。通过这种方式，我们可以将N表示为某个Banach空间的子空间。然后，我们将N中的其他映射引入到一些欧氏空间中，这些映射利用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离散拉普拉斯算子。我们引入了一种新的统计方法来求网络构型空间上离散拉普拉斯算子的平均值。通过这种方法，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与网的选择无关的极限；我们还证明了这在某种意义上与科威特-Machigashira-Shioya意义下的拉普拉斯算子一致。其次，我们通过比较不同空间和网的两个离散拉普拉斯算子，定义了A上的新结构。由于在信息几何中，两个分布的相对熵决定了雷曼度量，我们首先从拉普拉斯函数引入平稳马尔可夫链，然后将相对熵应用于它们，最后构造它们的连续极限，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.