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The study of relations between topological properties and differential geometric properties of foliated structures.

研究叶状结构的拓扑性质和微分几何性质之间的关系。

基本信息

批准号：
13640056
负责人：
OSHIKIRI Gen-ichi
金额：
$ 2.24万
依托单位：
Iwate University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

1) It is shown that codimension-one minimal foliation of a complete Riemannian manifold with non-negative Ricci curvature is totally geodesic if the growth of the foliation is not greater than 2. Further, an another proof of the estimate given by Miranda on the integral of the square norm of the second fundamental form of minimal graphs in Euclidean Spaces is obtained2) A kind of "Compact Leaf Theorem" of codimension-q metric foliations on closed Riemannian manifolds with positive curvature is obtained. As a corollary to this result, an extension of Berger's result on Killing fields is obtained : Any Killing field on a closed Riemannian manifolds with positive curvature has zero points or closed orbits.3) It is shown that Cheeger constant can be defined on (di-)graphs, and is related to connectivities of (di-)graphs.4) It is shown that the notion of admissible functions, which had already been defined for codimension-one foliations, can also be defined on digraphs, and that there is a strong relation between these two notions of "admissible functions" via the correspondence of a foliated manifold with the associated digraph. As an application, a divergence-like characterization of admissible functions of digraphs are obtained.

1) 证明了具有非负 Ricci 曲率的完全黎曼流形的余维一最小叶状结构，如果叶状结构的增长不大于 2，则它是完全测地线的。此外，还得到了 Miranda 对欧几里得空间中最小图的第二基本形式的平方范数积分的估计的另一个证明。 2) 一种“紧叶定理” 获得了具有正曲率的闭黎曼流形上的余维q度量叶状结构。作为这一结果的推论，得到了 Berger 在 Killing 场上的结果的扩展：具有正曲率的闭黎曼流形上的任何 Killing 场都具有零点或闭轨道。 3) 结果表明，Cheeger 常数可以在 (di-) 图上定义，并且与 (di-) 图的连通性有关。 4) 结果表明，已定义的容许函数的概念为余维一叶状结构，也可以在有向图上定义，并且通过叶状流形与相关有向图的对应关系，“允许函数”的这两个概念之间存在很强的关系。作为一个应用，获得了有向图的容许函数的类散度表征。