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Studies of glued Riemannian manifolds

Research Project

Project/Area Number 13640068
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

INNAMI Nobuhiro  Niigata University, Faculty of Sciences, Professor, 理学部, 教授 (20160145)

Co-Investigator(Kenkyū-buntansha) SEKIGAWA Kouei  Niigata University, Faculty of Sciences, Professor, 理学部, 教授 (60018661)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Keywordsgeometru of geodesics / Riemannian geometry / 平面凸ビリヤード / 測地線の幾何 / 凸ビリヤード問題 / リーマン多様体 / 張り合わせ多様体 / 測地線
Research Abstract

We find what rendition on gradient vector fields characterizes warped products, Riemannian products and round spheres. To do this we apply the theory of Jacobi equations without conjugate points to the differential maps of the local one-parameter groups generated by gradient vector fields.
We say that a manifold M is a glued manifold if M is a union of complete connected manifolds which are glued at their boundary Geodesies in a glued Riemannian manifold M are by definition locally minimizing curves in M. The variation vector fields through geodesies satisfy the Jacobi equation in each component manifold In this project we find the equation which show how Jacobi vector fields change in passing across the boundary of a component manifold into the neighboring component As an application we characterize glued Riemannian manifolds whose glued boundary separates conjugate points.
Circles and Ellipses has been characterized by some properties of billiard ball trajectories. Those properties have been discussed in connection with the characterization of flat metrics on tori by some families of geodesics and tori of revolution. The main method is the geometry of geodesies due to H.Busemann which was reconstructed in the configuration space by V.Bangert. In particular, the theory of parallels plays an important role in this work. Roughly speakin, there exists a foliation of parallels in the configuration space for billiards if and only if there exists a foliation of non-null homotopic curves in the phase space which is invariant under the billiard ball map.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (17 results)

All 2005 2004 2003 2002 2001 Other

All Journal Article (13 results) Publications (4 results)

  • [Journal Article] Notes on the Goldberg conjecture in dimension four.2005

    • Author(s)
      T.Oguro
    • Journal Title

      Complex, contact and symmetric manifolds, Progress in Mathematics 234

      Pages: 221-233

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Notes on strictly almost Kahler Einstein manifolds of dimension four2004

    • Author(s)
      T.Oguro
    • Journal Title

      Yokohama Math.J. 51

      Pages: 19-27

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Notes on strictly almost Kahler Einstein manifolds of dimension four2004

    • Author(s)
      T.Oguro, K.Sekigawa
    • Journal Title

      Yokohama Math.J. 51

      Pages: 19-27

    • NAID

      120001740816

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Notes on Kahler surfaces with distinct constant Ricci eigenvalues2003

    • Author(s)
      T.Nihonyanagi
    • Journal Title

      J.Korean Math.Soc. 40

      Pages: 1015-1029

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Notes on Kahler surfaces with distinct constant Ricci eigenvalues2003

    • Author(s)
      T.Nihonyanagi T.Oguro, K.Sekigawa
    • Journal Title

      J.Korean Math.Soc. 40

      Pages: 1015-1029

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Geometry of geodesics for convex billiards and circular billiards2002

    • Author(s)
      Nobuhiro Innami
    • Journal Title

      Nihonkai Math.J 13

      Pages: 73-120

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On 4-dimensional CR-submanifolds of a 6-dimensional sphere. Minimal surfaces2002

    • Author(s)
      H.Hashimoto
    • Journal Title

      Adv.Studies in Pure Math 34

      Pages: 143-154

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Geometry of geodesics for convex billiards and circular billiards2002

    • Author(s)
      Nobuhiro Innami
    • Journal Title

      Nihonkai Math.J. 13, 1

      Pages: 73-120

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On 4-dimensional CR-submanifolds of a 6-dimensional sphere. Minimal surfaces2002

    • Author(s)
      H.Hashimoto, K.Mashimo, K.Sekigawa
    • Journal Title

      Adv.Studies in Pure Math. 34

      Pages: 143-154

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Jacobi vector fields along geodesics in glued Riemannian manifolds2001

    • Author(s)
      Nobuhiro Innami
    • Journal Title

      Nihonkai Math.J. 12

      Pages: 101-112

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Gradient vector fields which characterize warped products2001

    • Author(s)
      Nobuhiro Innami
    • Journal Title

      Math.Scand. 88

      Pages: 182-192

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Gradient vector fields which characterize warped products2001

    • Author(s)
      Nobuhiro Innami
    • Journal Title

      Math.Scan. 88

      Pages: 182-192

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Jacobi vector fields along geodesies in glued Riemannian manifolds2001

    • Author(s)
      Nobuhiro Innami
    • Journal Title

      Nihonkai Math.J. 12, 2

      Pages: 101-112

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] INNAMI, Nobuhiro: "Geometry of geodesics for convex billiards and circular billiards"Nihonkai Math. J.. 13・1. 73-120 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] SEKIGWA, Kouei: "On 4-dimensional CR-submantfolds of a 6-dimensional sphere"Advanced Studies in Pure Mathematics. 34. 143-154 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Innami, Nobuhiro: "Gradient vector fields which characterize warped products"Math. Scand. 88・2. 182-192 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Innami, Nobuhiro: "Jacobi vector fields along geodesics in glued Riemannian manifolds"Nihonkai Math. J. 12・1. 101-112 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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