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2003 Fiscal Year Final Research Report Summary

Study on contac structures and foliations on 3 and 4 dimensional manifolds

Research Project

Project/Area Number 13440026
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

MITSUMATSU Yoshihiko  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70190725)

Co-Investigator(Kenkyū-buntansha) MIYOSHI Shigeaki  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60166212)
TSUBOI Takashi  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40114566)
SATO Hajime  Nagoya University, Graduate School of Mathematical Sciences, Professor, 大学院・多元数理科学研究科, 教授 (30011612)
TAKAKURA Tatsuru  Chuo University, Faculty of Science and Engineering, Associate Professor, 理工学部, 助教授 (30268974)
ONO Kaoru  Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20204232)
Project Period (FY) 2001 – 2003
Keywordscontact structures / foliations / symplectic structures / bi-contact structures / Ansov flows / foliated cohomology / asymptotic linking / Stein surface
Research Abstract

Based on the notion of asymptotic linking, the head investigator proposed the framework where the study on contact structures and that on foliations would be unified, and began the research. Op the side of foliations, it turned out that exotic classes and the 1st foliated cohomology are strongly related to this framework. On the other side, the torsion invariant of contact structures has a deep relation with it. For algebraic Anosov foliations, we also established the computation of its 1st foliated cohomology and found its relation to local orbit rigidity.
A research group including Miyoshi and Mitsumatsu investigated Thurston's inequality for foliations on the boundary of compact Stein surfaces and established the absolute version of the inequality for certain cases. The relative ersion and more general case are left for the future as important subject. Tsuboi and Mitsumatsu worked on the perfectness of groups of diffeomorphisms preservein certain geometric structures. Especially Tsuboi provednthe perfectness for contact diffeomorphisms and analytic diffeomorphisms of certain manifolds. Tsuboi also classified regular bi-contact structures on Seifert fibered spaces.
Another group including Ono and Ohta, mainly working on contact/symlectic topology, characterized the symplectic diffeo-types of the filling of the link of simple and hyper-elliptic singularities. Also they got started the construction of obstruction theory for Lagrangian Floer homology theory.

  • Research Products

    (46 results)

All Other

All Publications (46 results)

  • [Publications] Yoshihiko MITSUMATSU: "Foliations and contact structures on 3-manifolds"Proceedings of FOLIATIONS : GEOMETRY AND DYNAMICS in Warsaw, 2000. 75-125 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 三松 佳彦, 矢野 泰久: "Riemann多様体上の非圧縮流体の幾何"京都大学数理解析研究所講究録. 1260. 33-47 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigenori MATSUMOTO, Yoshihiko MITSUMATSU: "Leafwise cohomology and rigidity of certain Lie group actions"Ergodic Theory and Dynamical Systems. 23. 1839-1866 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Tetsuya Ozawa, Hajime SATO: "Contact transformations and their Schwarzian derivatives"Advanced Studies in Pure Math.. 37. 337-366 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hajime SATO, Tetsuya Ozawa: "Conformal Schwarzian Derivatives and Differential Equations"Proceeding of "Geometry, Integrability and Quantization", Coral Press. 4. 271-283 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 佐藤 肇: "2階偏微分方程式系の接触幾何学"数学. 55. 155-165 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takashi TSUBOI: "On the perfectness of groups of diffeomorphisms of the interval tangent to the identity at the endpoints"Proceedings of Foliations : Geometry and Dynamics, Warsaw 2000, World Scientific, Singapore. 421-440 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takeo NODA, Takashi TSUBOI: "Regular projectively Anosov flow without compact leaves"Proceedings of Foliations : Geometry and Dynamics, Warsaw 2000, World Scientific, Singapore. 403-419 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigenori MATSUMOTO, Takashi TSUBOI: "Transverse intersections of foliations in three-manifolds"Monographie de l'Enseignement Math\'ematique. 38. 503-525 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takashi TSUBOI: "Regular projectively Anosov flows on the Seifert fibered spaces"Journal of the Mathematical Society of Japan. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Tadayoshi MIZUTANI: "On Exact Poisson Manifolds of Dimension 3"Proceedings of Foliations : Geometry and Dynamics, Warsaw 2000, World Scientific, Singapore. 371-386 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Hagiwara, T.MIZUTANI: "Leibniz Algebras associated with foliations"Kodai Math.Journal. 25. 151-165 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeaki MIYOSHI: "A remark on torsion Euler classes of circle bundles"Tokyo Journal of Mathematics. 24. 189-194 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kenji Fukaya, Kaoru ONO: "Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds"Advanced Studies in Pure Mathematics. 31. 75-91 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kaoru ONO: "Space of geodesics on Zoll three-spheres"Advanced Studies in Pure Mathematics. 34. 237-247 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kaoru ONO: "Simple singularities and symplectic fillings"Contemporary Mathematics. 309. 195-197 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hiroshi OHTA, Kaoru ONO: "Symplectic fillings of links of simple elliptic singularities"Journal fur die Reine und Angewandte Mathematik. 565. 183-205 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hiroshi OHTA, Kaoru ONO: "Symplectic 4-manifolds containing singular rational curves with (2,3)-cusp"Congres et Seminaire, SMF. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 森吉 志: "非可換幾何学と指数定理"数学メモアール、日本数学会. 2. 157-228 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hiroshi OHTA: "Obstruction to and deformation of Lagrangian intersection Floer cohomology"Proccedings of the 4th KIAS Annual International Conference "Symplectic Geometry and Mirror Symmetry", ed.K.Fukaya et al.(World Scientific). 281-309 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hiroshi OHTA: "Brieskorn manifolds and metrics of positive scalar curvature"Advanced Studies in Pure Mathematics. 34. 231-236 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Tasuru TAKAKURA: "Hamiltonian group actions and equivariant indices"K-Monographs in Mathematics, Current Trends in Transformation Groups. 7. 217-229 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Tasuru TAKAKURA: "A note on the symplectic volume of the moduli space of spatial polygons"Advanced Studies in Pure Mathematics. 34. 255-259 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yoshihiko MITSUMATSU: "Foliations and contact structures on 3-manifolds"Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS in Warsaw. 2000. 75-125 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshihiko MITSUMATSU, Yasuhisa YANO: "Geometry of incompressible flow on Riemannian manifols"RIMS Proceedings. 1260. 33-47 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigenori MATSUMOTO, Yoshihiko MITSUMATSU: "Leafwise cohomology and rigidity of certain Lie group actions"Ergodic Theory and Dynamical Systems. 23. 1839-1866 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Tetsuya Ozawa, HaJime SATO: "Contact transformations and their Schwarzian derivatives"Advanced Studies in Pure Math.. 37. 337-366 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hajime SATO, Tetsuya Ozawa: "Conformal Schwarzian Derivatives and Differential Equations"Proceeding of "Geometry, Integrability and Quantization". 4. 271-283 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hajime SATO: "Contact geometry of partial differential systems of order 2"Sugaku Expositions. 55. 155-165 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Tsuboi: "On the perfectness of groups of diffeomorphisms of the interval tangent to the identity at the endpoints"Proceedings of Foliations: Geometry and Dynamics, Warsaw 2000, World Scientific, Singapore. 421-440 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takeo NODA, Takashi TSUBOI: "Regular projectively Anosov flow without compact leaves"Proceedings of Foliations: Geometry and Dynamics, Warsaw 2000, World Scientific, Singapore. 403-419 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigenori MATSUMOTO, Takashi TSUBOI: "Transverse intersections of foliations in three-manifolds"Monographie de l'Enseignement Math\'ematique. 38. 503-525 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi TSUBOI: "Regular projectively Anosov flows on the Seifert fibered spaces"Journal of the Mathematical Society of Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Tadayoshi MIZUTANI: "On Exact Poisson Manifolds of Dimension 3"Proceedings of Foliations: Geometry and Dynamics, Warsaw 2000, World Scientific, Singapore. 371-386 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Hagiwara, T.MIZUTANI: "Leibniz Algebras associated with foliations"Kodai Math. Journal. 25. 151-165 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeaki MIYOSHI: "A remark on torsion Euler classes of circle bundles"Tokyo Journal of Mathematics. 24. 189-194 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kenji Fukaya, Kaoru ONO: "Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds"Advanced Studies in Pure Mathematics. 31. 75-91 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kaoru ONO: "Space of geodesics on Zoll three-spheres"Advanced Studies in Pure Mathematics. 34. 237-247 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kaoru ONO: "Simple singularities and symplectic fillings"Contemporary Mathematics. 391. 195-197 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi OHTA, Kaoru ONO: "Symplectic fillings of links of simple elliptic singularities"Journal fur die Reine und Angewandte Mathematik. 565. 183-205 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi OHTA, Kaoru ONO: "Symplectic 4-manifolds containing singular rational curves with (2,3)-cusp"Congress et Seminaire, SMF. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hitoshi MORIYOSHI: "Non-commutative geometry and index theorem"Mathematics Memoir, the Mathematical Society of Japan. vol.2. 157-228 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi OHTA: "Obstruction to and deformation of Lagrangian intersection Floer cohomology"Proceedings of the 4th KIAS Annual International Conference" Symplectic Geometry and Mirror Symmetry" (ed. K.Fukaya et al.) (World Scientific). 281-309 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi OHTA: "Brieskorn maniforlds and metrics of positive scalar curvature"Advanced Studies in Pure Mathematics. 34. 231-236 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Tasuru TAKAKURA: "Hamiltonian group actions and equivariant indices"K-Monographs in Mathematics (Current Trends in Transformation Groups). vol.7. 217-229 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Tasuru TAKAKURA: "A note on the symplectic volume of the moduli space of spatial polygons"Advanced Studies in Pure Mathematics. 34. 255-259 (2001)

    • 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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