2003 Fiscal Year Final Research Report Summary
The study of relations between topological properties and differential geometric properties of foliated structures.
| Project/Area Number |
13640056
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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 | Iwate University |
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Principal Investigator |
OSHIKIRI Gen-ichi Iwate University, Faculty of Education, Professor, 教育学部, 教授 (70133931)
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| Co-Investigator(Kenkyū-buntansha) |
KAWADA Koichi Iwate University, Faculty of Education, Professor, 教育学部, 助教授 (70271830)
KOMIYAMA Haruo Iwate University, Faculty of Education, Professor, 教育学部, 助教授 (90042762)
KOJIMA Hasashi Iwate University, Faculty of Education, Professor, 教育学部, 教授 (90146118)
IIDA Masato Iwate University, Faculty of Education, Ass.Professor, 教育学部, 助教授 (00242264)
MIYAI Akio Iwate University, Faculty of Education, Lecturer, 教育学部, 講師 (70003960)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Foliation / Minimal foliation / Metric foliation / Killing field / Cheeger constant / (Di-)graph / Connectivities of graph / admissible function of digraph |
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| Research Abstract |
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 obtained
2) 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.
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