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1990 Fiscal Year Final Research Report Summary

Geometric Structures on Manifolds and Representations of Fundamental Group

Research Project

Project/Area Number 01540001
Research Category

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

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionDepartment of mathematics, Kumamoto University (1990)
Hokkaido University (1989)

Principal Investigator

KAMISHIMA Yoshinobu  Department of Mathematics, Kumamoto University, Associate Professor of Mathematics, 理学部, 助教授 (10125304)

Project Period (FY) 1989 – 1990
KeywordsCR-structure / Lorentz structure / Holonomy group / Fundamental group / Spherical CR structure / Conformally flat structure / One parameter group / Topological rigidity
Research Abstract

A CR-structure on a 2n + 1 dimensional smooth manifold M consists of a pair (omega, J) satisfying that (i) omega is a contact form of M , and (ii) Let Null omega = {X * TM* omega (X) = 0} which is a codimension 1 sub-bundle of the tangent bundle TM. Then there is a complex structure J on Null omega. Namely J is an almost complex structure on Null omega and when Null omega <cross product> C = T^<1,0> + T^<0,1> is the canonical splitting, it follows that [T^<1,0>, T^<1,0>] * T^<1,0>. In addition a CR-structure is strictly pseudo-convex if the Hermitian pairing Q : T^<10>XT^<1,0> -> C defined by Q (X, Y) = d-omega (X, JY) is positive definite. In 1958 Boothby and Wang introduced the regular contact structure on smooth manifolds and they have established the vibration theorem. A regular CR-structure will be defined as a CR-structure whose underlying contact structure is regular. Then the Boothby and Wang's result will be generalized as follows : M^<2n+1> admits a regular CR-structure if and only if M is a principal circle bundle pi : M -> N over a Kahler manifold N whose fundamental 2-form OMEGA satisfies the following properties ; (1) the euler class of the bundle is represented by an integral cocycle [OMEGA] * H^2 (N ; Z). (2) deta = pi^*OMEGA where eta is a connection form of M.

  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] 神島 芳宣: "Conformal Cincle actuis an 3ーmanihlds" Sprihgn Lectes Notes in natl. 1375. 132-144 (1989)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 神島 芳宣: "Lorantr structues and Killiny vectr filds on manihels" Proceedruys of Wurkshops in Pvner Matl. 9. 75-85 (1990)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 神島 芳宣: "Conformal wutomuphius and wlalytlut" Trans,A,M,S.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 神島 芳宣: "CRーstructunes on Seifent monifolcs" Invent.Math.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yoshinobu Kamishima: "Conformal circle actions on 3-manifolds" "Transformation Groups, Proceedings, Osaka University, 1987-88". Springer Lecture Notes in Math. 1375. 132-144 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinobu Kamishima: "Lorentz structures and Killing vector fields on manifolds" "Topology and Geometry of Manifolds, Proceedings, Pohang Institute of Science and Technology in Korea, 1989". Proceedings of Workshops in Pure Math. 9. 75-85 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinobu Kamishima: "Conformal automorphisms and conformally flat manifolds" Trans. Amer. Math. Soc.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinobu kamishima: "CR-structures on Seifert manifolds" Invent. Math.

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

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Published: 1993-08-12  

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