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Foliations and Geometric Structures

Research Project

Project/Area Number 02640015
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionChiba Universtiy

Principal Investigator

INABA Takashi  Chiba University, College of Arts and Sciences, Assistant Professor, 教養部, 助教授 (40125901)

Co-Investigator(Kenkyū-buntansha) YASUDA Masami  Chiba University, College of Arts and Sciences, Professor, 教養部, 教授 (00041244)
HINO Toshiyuki  Chiba University, College of Arts and Sciences, Professor, 教養部, 教授 (70004405)
ANDO Tetsuya  Chiba University, College of Arts and Sciences, Assistant Profeesor, 教養部, 助教授 (20184319)
NOZAWA Sohei  Chiba University, College of Arts and Sciences, Assistant Profeesor, 教養部, 助教授 (20092083)
KUGA Ken'ichi  Chiba University, College of Arts and Sciences, Assistant Professor, 教養部, 助教授 (30186374)
Project Period (FY) 1990 – 1991
Project Status Completed (Fiscal Year 1991)
Budget Amount *help
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1991: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 1990: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsFoliation / Geometric Structure / Global Holonomy Group / 接アフィン葉層 / レヴィ葉層 / 接アフィン関数 / 弾性葉 / 例外葉
Research Abstract

There are two natural ways to introduce geometric structures into foliations. One is to introduce them in the direction transverse to the leaves, and the other is to do so in the direction tangent to the leaves. The former way has already been studied very much by many people, but it seems that works about the latter way are not so many in the literature.
In this research, we investigated these both types of geometric structures on foliations, mainly from the viewpoint of differential topology. Firstly, as for the transverse geometric structures, we studied co-dimension one foliations with transverse projective structure. We clarified the relation between the global holonomy group and the topological properties of the foliations. Furthermore, we showed that a transversely projective foliation cannot have exceptional leaves if the ambient manifold has amenable fundamental group. Secondly, as for the tangential geometric structures, we studied(1)tangentially affine foliations and(2)tangentially holomorphic foliations : (1)Note that tangentially affine foliations appear as Lagrangian foliations on symplectic manifolds. We determined the space of all leafwise affine, functions on a tangentially affine foliation on the torus. We also proved that the three dimensional sphere does not admit any co-dimension one tangentially affine foliation. (2)A Levi-flat real hypersurface in a complex surface has a tangentially holomorphic foliation, which is usually called the Levi foliation. We obtained some topological properties of compact Levi-flat hypersurfaces by investigating the holonomy of total leaves in their Levi foliations. In particular, we showed that the three dimensional sphere cannot be embedded as a Levi-flat hypersurface. This result is applied to two dimensional complex dynamical systems.

Report

(3 results)
  • 1991 Annual Research Report   Final Research Report Summary
  • 1990 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Takashi Inaba and Shigenori Matsumoto: "Resilient leaves in transversely projective foliations" J.Fac.Sci.Univ.Tokyo. 37. 89-101 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Takashi Inaba and Shigenori Matsumoto: "Nonsigular expansive flows on 3-menifolds and foliations with circle provy singulatities" Japan.J.Math.16. 329-340 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] D.E.Barrett and Takashi Inaba: "On the topology o compact smooth three dimensinal Levi-flat bypersurfaces"

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Takashi Inaba and Kazuo Masuda: "Tanycutially a ffine foliations and leafwise a ffine functione an the forus"

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Ken'ichi Kuga: "Certain polynomials for knots with integral representations"

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Tetsuya Ando: "On the normal bundle of P' in the higher dimeusinal projective variety" American J. Math.113. 949-961 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Takashi Inaba and Shigenori Matsumoto: "Resilient Leaves in Transversely Projective Foliations" J. Fac. Sci. Univ. Tokyo. 37. 89-101 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Takashi Inaba and Shigenori Matsumoto: "Nonsingular Expansive Flows on 3-Manifolds and Foliations with Circle Prong Singularitier" Japan. J. Math.16. 329-340 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] D. E. Barrett and Takashi Inaba: "On the TOPOLOGY of Compact Smooth Three Dimensional Levi-Flat Hypersurfaces"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Takashi Inaba and Kazuo Masuda: "Tangentially Affine Foliations and Leafwise Affine Functions on the Torus"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Ken'ichi Kuga: "Certain Polynomials for Knots with Integral Representations"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Tetsuya Ando: "On the Normal Bundle of P1 in the Higher Dimensional Projectrive Variety" Amer. J. Math.113. 949-961 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Takashi Inaba and Nobuo Tsuchiya: "Expansive Foliations" Hokkaido Math.J.21. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] D.E.Barrett and Takashi Inaba: "On the topology of compact smooth three dimensional Leviーflat hypersurfaces"

    • Related Report
      1991 Annual Research Report
  • [Publications] Takashi Inaba and Kazuo Masuda: "Tangentially affine foliations and leafwise affine functions on the torus"

    • Related Report
      1991 Annual Research Report
  • [Publications] Ken'ichi Kuga: "Certain polynomial for knots with integral representations"

    • Related Report
      1991 Annual Research Report
  • [Publications] Sohei Nozawa: "Sharp characters of finite qroups having precribed values" Tsukuba J.Math.

    • Related Report
      1991 Annual Research Report
  • [Publications] Yoshiyuki Hino and Satoru Murakami: "Total stability and umiformly asymptofic stability for lineor Volterra equations" J.Londm Math.Soc.43. 305-312 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Takashi Inaba and Shigenori Matsumoto: "Resilient leaves in transversely projective foliations" J.Fac.Sci.Univ.Tokyo. 37. 89-101 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Takashi Inaba and Shigenori Matsumoto: "Nonsingular expansive flows on 3ーmemfolds and foliations with circle prong singularities" Japan.J.Math.16. 329-340 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Ken'ichi Kuga: "A Note on Lipschitz distance for smooth structures on nonーcompact mamfolds" J.College of Arts and Sci.Bー23. (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Sohei Nozawa: "On groups with a selfーcentralizing Sylow pーsubgroup" J.College of Arts and Sci.Bー23. (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Tetsuya Ando: "On the normal bundle of P^1 in the higher clinensianel projective variety" Amer.J.Math.

    • Related Report
      1990 Annual Research Report
  • [Publications] Yoshiyuki Hino and Satoru Murakami: "Stabilidy properties of Lineen Volterra eguations" J.Diff.Eg.89. 121-137 (1991)

    • Related Report
      1990 Annual Research Report

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Published: 1990-04-01   Modified: 2016-04-21  

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