2006 Fiscal Year Final Research Report Summary
Topology of nonintegrable plane fields
| Project/Area Number |
16540053
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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 | Chiba University |
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Principal Investigator |
INABA Takashi Chiba University, Graduate School of Science and Technology, Professor, 自然科学研究科, 教授 (40125901)
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| Co-Investigator(Kenkyū-buntansha) |
TSUBOI Takashi The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理科学研究科, 教授 (40114566)
KUGA Ken'ichi Chiba University, Faculty of Science, Professor, 理学部, 教授 (30186374)
HINO Yoshiyuki Chiba University, Faculty of Science, Professor, 理学部, 教授 (70004405)
TAKAGI Ryoichi Chiba University, Faculty of Science, Professor, 理学部, 教授 (00015562)
SUGIYAMA Ken-ichi Chiba University, Faculty of Science, Associate Professor, 理学部, 助教授 (90206441)
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| Project Period (FY) |
2004 – 2006
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| Keywords | nonintegrable plane fields / Engel structure / rigidity / characteristic curve / projective structure / contact manifolds of higher order / geometric entropy |
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| Research Abstract |
The purpose of this research was to study nonintegrable plane fields from the topological viewpoint and clarify their global behavior.
First, we considered the rigidity of loops tangent to Engel plane fields. Given a characteristic curve with the initial point being fixed, we completely determined how the terminal point of the curve can vary by small perturbations of the curve in the space of tangential curves to the plane field. Especially, we obtained the following: The trace of terminal points under perturbations becomes an open set if and only if the developing image of the curve with respect to the canonical projective structure coincides with the whole projective line. As an application of this result we got the following: Any non-affine characteristic loop is non-rigid. We also showed that every 1-dimensional projective structure of the circle can be realized as the canonical projective structure of some characteristic loop in some Engel manifold.
Next, we studied the rigidity in higher dimensions. We showed that maximal integral submanifolds of the symbol plane fields on contact manifolds of higher orders are always locally rigid. We also produced an example of a rigid torus in some manifold endowed with a nonintegrable plane field.
Thirdly, we tried to generalize the Ghys-Langevin-Walczak geometric entropy of foliations to the nonintegrable cases. To define an entropy, we need to use integral curves. We recognized that if we exclusively use integral curves with bounded geometry we are able to define a notion of entropy for nonintegrable plane fields.
Parts of these results have been published in the proceedings of the international conference FOLIATIONS 2005, under the title : On rigidity of submanifold a tangent to nonintegrable foliations.
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