2002 Fiscal Year Final Research Report Summary
Geometry of space of Riemannian manifolds
| Project/Area Number |
11640075
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kyushu University (2001-2002)
Osaka University (1999-2000) |
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Principal Investigator |
OTSU Yukio Kyushu University, Department of Mathematical Sciences, Ass. Prof., 大学院・数理学研究院, 助教授 (80233170)
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| Co-Investigator(Kenkyū-buntansha) |
GOTO Ryushi Osaka University, Department of Sciences, Ass. Prof., 大学院・理学研究科, 助教授 (30252571)
SHIOYA Takashi Tohoku University, Department of Sciences, Ass. Prof., 大学院・理学研究科, 助教授 (90235507)
YAMADA Koutarou Kyushu University, Department of Mathematical Sciences, Prof., 大学院・数理学研究院, 教授 (10221657)
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| Project Period (FY) |
1999 – 2002
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| Keywords | Alexandrov space / Hausdorff distance / comparions geometry / Laplacian / Convergence theorem |
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| Research Abstract |
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.
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