Geometry of space of Riemannian manifolds

下面用A表示有界曲率的Alexnadrov空间，上面用具有Hausdorff距离的Hausdorff维空间，用I表示A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。由于所有网的构形空间被标识为空间的乘积，因此A及其网中空间对的集合N可以理解为A上的一个光纤空间。我们考虑一个映射，该映射分配了每个网两点的互距离矩阵。这样我们就可以把N表示成某个巴拿赫空间的子空间。然后我们引入从N到某个欧几里德空间的其他映射，这些映射取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函数的离散拉普拉斯式。提出了一种新的统计方法来求网络位形空间上离散拉普拉斯算子的平均。通过这种方法，我们证明了离散拉普拉斯函数的特征值和特征向量收敛于极限，与网络的选择无关；我们也在某种程度上证明了它与kuwai - machigashira -shioya的拉普拉斯定理是一致的。接下来，由于它们是矩阵空间中的同一成员，我们通过比较不同空间和网的两个离散拉普拉斯算子来定义A上的新结构。由于信息几何中两个分布的相对熵决定了黎曼度规，我们首先从拉普拉斯矩阵中引入平稳马尔可夫链，然后对它们应用相对熵；最后构造了它们的连续极限，这是对Hausdorff距离的一种推广。

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.