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2003 Fiscal Year Final Research Report Summary

Topological study of Engel structures and its characteristic foliations

Research Project

Project/Area Number 14540064
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionChiba University

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)

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)
Project Period (FY) 2002 – 2003
KeywordsEngel structure / Characteristic foliation / Transverse contact structure / Tangential projective structure / Rigid curve
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.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Takashi Inaba: "Open Engel manifolds admitting compact characteristic leaves"Bull.Australian Math.Soc.. 68. 213-219 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Inaba, H.Nakayama: "Invariant fiber measures of angular flows and the Rue11 invariant"J.Math.Soc.Japan. 56. 17-29 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takeo Noda, Takashi Tsuboi: "Regular projectively Anosov flows without compact leaves"Foliations : geometry and dynamics, World Sci.. 403-419 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yoshiyuki Hino, Satoru Murakami: "Almost automorphic solutions for abstract functional differential equations"J.Math.Anal.Appl.. 286. 741-752 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Daijiro Fukuda, Ken'ichi Kuga: "Twisted quantum doubles"International J.Math.Math.Sci.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shin Satoh: "Surface diagrams of twist-spun 2-knots"J.Knot Theory Ramifications. 11. 413-430 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takashi Inaba: "Open Engel manifolds admitting compact characteristic leaves"Bull.Australian Math.Soc.. 68-2. 213-219 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Inaba, Hiromichi Nakayama: "Invariant fiber measures of angular flows and the Ruell invariant"J.Math.Soc. Japan. 56-1. 17-29 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takeo Noda, Takashi Tsuboi: "Regular projectively Anosov flows without compact leaves"Foliations : geometry and dynamics, World Sci.. 403-419 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshiyuki Hino, Satoru Murakami: "Almost automorphic solutions for abstract functional differential equations"J.Math.Anal.Appl.. 286. 741-752 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Daijiro Fukuda, Ken'ichi Kuga: "Twisted quantum doubles"International J.Math Math Sci.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shin Satoh: "Surface diagrams of twist-spun 2-knots"J.Knot Theory Ramifications. 11. 413-430 (2002)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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