| Project/Area Number |
08640077
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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 | HOKKAIDO UNIVERSITY |
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Principal Investigator |
NAKAI Isao Hokkaido University Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90207704)
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| Co-Investigator(Kenkyū-buntansha) |
MINAKAWA Hiroyuki Hokkaido University Graduate School of Science. Instructor, 大学院・理学研究科, 講師 (30241300)
SUWA Tasuo Hokkaido University Graduate School of Science.Professor, 大学院・理学研究科, 教授 (40109418)
SATO Hajime Nagoya University Graduate School of Mathematic, Profesor, 多元数理科学研究所, 教授 (30011612)
KAWAZUMI Nariya Hokkaido University Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30214646)
ISHIKAWA Goo Hokkaido University Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50176161)
泉屋 周一 北海道大学, 大学院・理学研究科, 教授 (80127422)
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| Project Period (FY) |
1996 – 1997
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| Project Status |
Completed (Fiscal Year 1997)
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| Budget Amount *help |
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 1997: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1996: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Web / Foliation / Chern onnection / Integrable system / Complex dynamics / Non solvable Dynamics / First Order PDE / Higher Order ODE / 高階偏微分方程式 / Web / 一階微分方程式 / ルジャンドル / 接触構造 / pseudo group / 留数公式 |
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| Research Abstract |
Differential geometry of Web structure is to study the relation of geometric structure of configurations of foliations and its affine connection. In general the connection is not unique but there are finitely many connections. In this research project, I determined all configurations of codimension one foliations for which all those connections are equal. This result generalizes the classical result due to Poincare, Reidemeister and Mayrhofer. And also I showed that in general the mean of curvature forms of all those affine connections is the curvature form defined by Blaschke in 1930's. This observation motivated to apply Web geometry to certain integrable systems. A holonomic partial differential equation on R' is a n-dimensional variety in its projective cotangent bundle. On the variety the contact form restricts to one form, of which the integrable manifolds are the solutions of the equations. Web geometry applies here to extend Bott connection of the foliation by the solutions to unique affine connection on the variety. One of my results tells the mean of the resulting affine connection projected to the base space R' gives Blaschke curvature form. For a certain moduli space of holonomic PDE with complete integrals I showed it is in one to one correspondence with the space of Blaschke curvature forms. This result was announced in the symposium on Web geometry "Journees sur les Tissus" held at Univ. Paul Sabatier, Toulouse in 1996 December, and the paper on the result is in preparation to publish as a part of the lecture note of the symposium. In another vean, I discussed the local problem of the classification problem of the domains in the complex plane with analytic boundaries by using a method in complex dynamical systems.
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