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2001 Fiscal Year Final Research Report Summary

DIFFERENTIAL GEOMETRIC RESEARCH ON SUBMANIFOLDS

Research Project

Project/Area Number 11440024
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTOKYO METROPOLITAN UNIVERSITY

Principal Investigator

OGIUE Koichi  GRADUATE SCHOOL OF SCIENCE, DEPARTMENT OF MATHEMATICS, PROFESSOR, 理学研究科, 教授 (10087025)

Co-Investigator(Kenkyū-buntansha) KOISO Miyuki  DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY OF EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (10178189)
MAEDA Sadahiro  DEPARTMENT OF MATHEMATICS, SHIMANE UNIVERSITY, PROFESSOR, 総合理工学部, 教授 (40181581)
OHNITA Yoshihiro  GRADUATE SCHOOL OF SCIENCE, DEPARTMENT OF MATHEMATICS, PROFESSOR, 理学研究科, 教授 (90183764)
MIYAOKA Reiko  DEPARTMENT OF MATHEMATICS, SOPHIA UNIVERSITY, PROFESSOR, 理工学部, 教授 (70108182)
HAMADA Tatsuyoshi  DEPARTMENT OF APPLIED MATHEMATICS, FACULTY OF SCIENCE, FUKUOKA UNIVERSITY, RESEARCH ASSOCIATE, 理学部, 助手 (90299537)
Project Period (FY) 1999 – 2001
KeywordsSubmanifold / Differential Geometry / Curve / Surface / Curvature / Riemannian manifold / Geometric structure
Research Abstract

In this project, we had several sizes of symposium, international conferences, workshops etc. 3 times in 1999, 3 times in 2000, 2 times in 2001. Especially, we organized the 46-th Geometry Symposium at Univ. of Tokyo in August, 1999, the 9-th MSJ-IRI " Integrable Systems in Differential Geometry " in July, 2000, and an international workshop on Geometry of Riemannian Submanifolds at Tokyo Metropolitan Univ. in December, 2001.
There we were able to have talks of results, discussions and interchanges of informations about differential geometric research on submanifolds synthetically and to receive the review for this project. Ogiue, Maeda, Adachi developed study of curves, in particular circles, in more general symmetric spaces and their submanifolds than complex space forms, and obtain the results. Especially their research examining in details the length spectrum of geodesic spheres in complex space froms by the number theoretic method were highly estimated and published in J. Math. Soc … More . Japan. Miyaoka gave a different proof of homogeneity for the case six principal curvatures and multiplicity 1 in the classification problem of isoparametric hypersurafces and her work got the high estimate. Further in cooperation with Ishikawa, and Kimura, she constructed many compact submanifolds in spheres with degenerate Gauss maps satisfying the equality of Ferus and gave many examples of special Lagrangian submanifolds as a by-product. Kitagawa studied isometric deformations of flat tori in 3-dimensional standard sphere with nonconstant mean curvature and proved that these flat tori are all deformable. Kenmotsu classified parallel mean curvature surfaces with constant Gaussian curvature in complex space forms by analyzing precisely a certain nonlinear ordinary differential equation and his work was decided to be published in an internationally high journal. In joint works with Hashimoto and Mashimo, Sekigawa gave the structure equations for 4-dimensional CR-submanifolds in the nearly K\"ahler 6-dimensional sphere by the moving frame method in Bryant's style, and as applications he studied fundamental properties of characteristic classes of such submanifolds and provided new examples by the Lie theoretic method. Koiso analyzed actively deformation and stability problem of constant mean curvature surfaces with boundary and obtained excellent results. On the other hand, Ogiue, Nakamula, Hamada greatly have maken good use of the system (PPDG) interchanging extensively and effectively research results and informations, and we expect that this system will be developed to the Geometry Server in future by Guest and Ohnita. Less

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] S.Maeda: "Length spectrum of geodesic spheres in a non-flat complex space form"J.Math.Soc.Japan. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Maeda: "A characterization of the second Veronese embedding into a complex projective space"Proc.Japan Acad.. 77. 99-102 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Maeda: "Hopf hypersufaces with constant principal curvatures in complex projective or complex hyperbolic spaces"Tokyo J.Math.. 24. 133-152 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Koiso: "Deformation and stability of surfaces with constant mean curvature"Tohoku Math.J.. 54. (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Sekigawa: "On some four-dimensional almost Kahler Einstein manifolds"Kodai Math.J.. 24. 226-258 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Kitagawa: "Isometric deformations of flat tori in the 3-sphere with nonconstant mean curvature"Tohoku Math.J.. 52. 283-298 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Maeda: "Length spectrum of geodesic spheres in a non-flat complex space form"J.Math.Soc.Japan. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Maeda: "A characterization of the second Veronese embedding into a complex projective space"Proc. Japan Acad.. 77. 99-102 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Maeda: "Hopf hypersurfaces with constant principal curvatures into complex projective or complex hyperbolic spaces"Tokyo J.Math.. 24. 133-152 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Koiso: "Deformation and stability of surfaces with constant mean curvature"Tohoku Math.J.. to appear, 54. (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Sekigawa: "On some four-dimensional almost K\" ahler Einstein manifolds"Kodai Math.J.. 24. 226-258 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Kitagawa: "Isometric deformations of flat tori in the 3-sphere with nonconstant mean curvature"Tohoku Math.J.. 52. 283-298 (2000)

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

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Published: 2004-04-14  

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