2003 Fiscal Year Final Research Report Summary
Topological study of Engel structures and its characteristic foliations
| Project/Area Number |
14540064
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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, 大学院・自然科学研究科, 教授 (40125901)
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| Co-Investigator(Kenkyū-buntansha) |
HINO Yoshiyuki Chiba University, Faculty of Science, Professor, 理学部, 教授 (70004405)
KUGA Ken'ichi Chiba University, Faculty of Science, Professor, 理学部, 教授 (30186374)
TSUBOI Takashi Univ.of Tokyo, Graduate School of Math Sci., Professor, 大学院・数理科学研究科, 教授 (40114566)
SATOH Shin Chiba University, Graduate School of Science and Technology, Assistant, 大学院・自然科学研究科, 助手 (90345009)
TAKAGI Ryoichi Chiba University, Faculty of Science, Professor, 理学部, 教授 (00015562)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Engel structure / Characteristic foliation / Transverse contact structure / Tangential projective structure / Rigid curve |
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| Research Abstract |
An Engel structure is a maximally non-integrable 2-dimensional plane field on a 4-dimensional manifold. In this research we investigated global properties of Engel structures from the viewpoint of topology. In particular, we paid attention to the characteristic 1-dimensional foliations canonically associated to Engel structures. First, we considered the following problem posed by Gershkovich : Does there exist an Engel structure on the 4-dimensional Euclidean space whose characteristic foliation admits a compact leaf ? We affirmatively solved this problem by constructing a concrete example. We also applied the construction to obtaining Engel structures on other open 4-manifolds. This result was published in Bulletin of the Australian Mathematical Society. Next, P.Walczak, the foreign joint investigator of this research, constructed an Engel structure using an Anosov flow. The head investigator remarked that this Engel structure is essentially new. Namely, it is not isotopic to any other known examples. Thirdly, we sought 1-dimensional transversely parallelizable foliations on 4-dimensional manifolds which cannot be topologically conjugate to the characteristic foliation of any Engel structure. It is known that a characteristic foliation admits a tangential projective structure and transverse contact structure. We observed the following fact : If a compact leaf of a characteristic foliation has finite holonomy, then the projective. structure of the leaf is not affine. Using this fact we found a l-dimensional transversely parallelizable foliation which is not topologically conjugate to the characteristic foliation of any Engel structure. Finally, we initiated the study of generalizing the rigidity property of characteristic foliations of Engel structures to the cases of higher dimensional plane fields.
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Research Products
(12 results)
