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2004 Fiscal Year Final Research Report Summary

Comparison on bow-shapes for Kaehler magnetic fields

Research Project

Project/Area Number 14540075
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNagoya Institute of Technology

Principal Investigator

ADACHI Toshiaki  Nagoya Institute of Technology, Graduate School of Engineering, 工学研究科, 教授 (60191855)

Co-Investigator(Kenkyū-buntansha) MAEDA Sadahiro  Shimane University, Interdisciplinary Faculty of Science and Engineering, 総合理工学部, 教授 (40181581)
EJIRI Norio  Nagoya Institute of Technology, Graduate School of Engineering, 工学研究科, 教授 (80145656)
UDAGAWA Seiichi  Nihon University, School of Medicine, 医学部, 講師 (70193878)
YAMAGISHI Masakazu  Nagoya Institute of Technology, Graduate School of Engineering, 工学研究科, 助教授 (40270996)
SAEKI Akihiro  Nagoya Institute of Technology, Graduate School of Engineering, 工学研究科, 助教授 (50270997)
Project Period (FY) 2002 – 2004
KeywordsKaehler magnetic fields / crescent / trajectories / Jacobi fields / magnetic exponential maps / sectors / comparison theorem / magnetic flow
Research Abstract

The head of investigator mainly studied real 2-dimensional surfaces which are naturally obtained from trajectories for Kaehler magnetic fields. He compared such surfaces on complex space forms with such surfaces on general Kaehler manifolds..
(1)Comparison on bow-shapes and crescents
Given a trajectory for a Kaehler magnetic field, we attach at each point on this trajectory a geodesic whose initial vector and the tangent vector of trajectory span a complex line in the tangent space. Such geodesics form a real 2-dimensional surfece. On a complex space form this surface is a complex space line. In order to study tie shape of this surfece on a general Kaehler manifold, for a part of this trajectory we take a geodesic segment on this surfece which joins its ends. Under the condition of sectional curvature from above, we find the length of the geodesic segment is not shorter than the length of bow-shape on a complex space form. We also find that the equality holds if and only if the crescent … More is totally geodesic immersed bow-shape.
(2)Comparison on sectors
We consider an image of a complex line through magnetic exponential map. On a complex space form the image of r-ball in a tangent complex line is an intersection of geodesic r-ball and a complex space line. In order to study this sector on a general Kaehler manifold, we consider the length of arc of this sector. Under the condition of sectional curvature from below, we find the length of the arc is not longer than the length of arc of a corresponding sector on a complex space form. We also find that the equality holds if and only if the sector is totally geodesic and complex embedded one.
(3)Variation of force and pseudo-congruency of Kaehler magnetic flows
Though we have a real 2-dimensional surface which is obtained by varying forces of Kaehler magnetic fields, it is not so natural on a general Kaehler manifold in view of mappings. Using this property we studied pseudo-congruency of Kaehler magnetic flows on a Kaehler manifold with complex Euclidean factor. As an application we characterize the rank of Hermitian symmetric space of noncompact type in terms of force of Kaehler magnetic field with horocycle trajectories. Less

  • Research Products

    (25 results)

All 2004 2003 2002 Other

All Journal Article (25 results)

  • [Journal Article] Congruence theorem of geodesics on some naturally reductive Riemannian homogeneous manifolds2004

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Mathematical Reports of the Academy of Science of the Royal Society of Canada 26

      Pages: 11-17

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Shape orerators and structure tensors of real hypersurfaces in nonflat quaternionic space forms2004

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Bulletin of the Polish Academy of Sciences Mathematics 52

      Pages: 319-328

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 階数1の対称空間における曲線と部分多様体2004

    • Author(s)
      足立俊明
    • Journal Title

      数学 56

      Pages: 33-48

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Congruence theorem of geodesies on some naturally reductive Riemannian homogeneous manifolds2004

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Mathematical Reports of the Academy of Science of the Royal Society of Canada 28

      Pages: 11-17

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Curves and submanifolds in a symmetric space of rank one(in Japanese)2004

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Suugaku 56

      Pages: 33-48

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Quaternionic distribution of curvature-adapted real Hypersurfaces in a quaternionic hyperbolic space2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Journal of Geometry 75

      Pages: 1-14

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Kaehler magnetic fields on a product of complex hyperbolic spaces2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Trends in complex analysis, differential geometry and mathematical physics

      Pages: 1-11

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Circles and Kaehler immersions2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Trends in complex analysis, differential geometry and mathematical physics

      Pages: 123-134

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Lamination of the moduli space of circles and their length spectrum for a non-flat complex space form2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Osaka Journal of Mathematics 40

      Pages: 895-916

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characterization of complex space forms in terms of geodesics on their geodesic spheres2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Nihonkai Journal of Mathematics 14

      Pages: 95-98

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Comparison theorems associated with Kahler magnetic fields2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Proceedings of the International Symposium on Analysis, manifolds and mechanics

      Pages: 20-27

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Circles and hypersurfaces in space forms2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Memoirs of the Faculty of Science and Engineering Shimane University, Series B : Mathematical Science 36

      Pages: 1-9

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Geometry of ordinary helices in a complex projective space2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Hokkaido Mathematical Journal 33

      Pages: 233-246

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Real hypersurfaces in nonflat complex space forms are irreducible2003

    • Author(s)
      Sadahiro MAEDA
    • Journal Title

      Osaka Journal of Mathematics 40

      Pages: 121-138

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A characterization of constant isotropic immersions by circles2003

    • Author(s)
      Sadahiro MAEDA
    • Journal Title

      Archiv der Mathematik 81

      Pages: 90-95

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characterization of complex space forms in terms of geodesies on their geodesic spheres2003

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Nihonkai Journal of Mathematics 14

      Pages: 95-98

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Length spectrum of geodesic spheres in a non-flat complex space form2002

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Journal of the Mathematical Society of Japan 54

      Pages: 373-408

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characterization of totally geodesic Kaehler immersions2002

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Hokkaido Mathematical Journal 91

      Pages: 629-641

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A characterization of the Veronese imbedding into a complex projective space2002

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Mathematical Reports of the Academy of Science of the Royal Society of Canada 24

      Pages: 61-66

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Study of submanifolds by curves of order 22002

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Memoirs of the Faculty of Science and Engineering Shimane University, Series B : Mathematical Science 35

      Pages: 1-21

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A Differential-Geometric Schottoky Problem and Minimal Surfaces in Tori2002

    • Author(s)
      Norio EJIRI
    • Journal Title

      Contemporary Mathematics 308

      Pages: 101-144

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A characterization of extrinsic spheres in a Riemannian manifold2002

    • Author(s)
      Sadahiro MAEDA
    • Journal Title

      Tsukuba Journal of Mathematics 26

      Pages: 291-297

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Three theorems on isotropic immersions2002

    • Author(s)
      Sadahiro MAEDA
    • Journal Title

      Trends in complex analysis, differential geometry and mathematical physics

      Pages: 116-122

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A characterization of the Veronese imbedding into a complex projective space

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Mathematical Reports of the Academy of Science of the Royal Society of Canada

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Comparison theorems associated with K" ahler magnetic fields

    • Author(s)
      Toshiaki ADACHI
    • Journal Title

      Proceedings of the International Symposium on Analysis, manifolds and mechanics

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

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Published: 2006-07-11  

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