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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)
北川 義久  宇都宮大学, 教育学部, 助教授 (20144917)
神島 芳宣  東京都立大学, 大学院・理学研究科, 教授 (10125304)
山田 光太郎  九州大学, 大学院・数理学研究科, 教授 (10221657)
剱持 勝衛  東北大学, 大学院・理学研究科, 教授 (60004404)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥15,100,000 (Direct Cost: ¥15,100,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2000: ¥6,100,000 (Direct Cost: ¥6,100,000)
Fiscal Year 1999: ¥7,500,000 (Direct Cost: ¥7,500,000)
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

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (31 results)

All Other

All Publications (31 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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Maeda: "Length spectrum of geodesic spheres in a non-flat complex space form"J. Math. Soc. Japan. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Maeda: "A characterization of the second Veronese embedding into a complex projective space"Proc. Japan Acad.. 77. 99-102 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Maeda: "Hopf hypersurfaces with constant principal curvatures in complex projective or complex hyperbolic spaces"Tokyo J. Math.. 24. 133-152 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Koiso: "Deformation and stability of surfaces with constant mean curvature"Tohoku Math. M.. 54(発行予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Ohnita: "On Hamiltonian stability of certain H-minimal Lagrangian submanifolds in Hermitian symmetric spaces"RIMS Kokyuroku, Geometry of Submanifolds and Related Topics. 1236. 31-48 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Sekigawa: "On some four-dimensional almost Kahler Einstein manifolds"Kodai Math. J.. 24. 226-258 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 前田定広: "Length spectrum of geodesic spheres in a non-flat complex space form"J.Math.Soc.Japan. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 剱持勝衛: "On minimal surfaces with constant curvature in two-dimensional complex space forms"J.Reine Angew.Math.. 523. 69-101 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 剱持勝衛: "The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms"Ames.J.Math.. 122. 295-317 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 山田光太郎: "General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux"Commun.in Analysis and Geometry. 8. 83-114 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 宮岡礼子: "Submanifolds with degenerate Gauss mappings in spheres"Proceedings of the first international symposium on differential geometry, Josai Math.Monogr.. No.3. 125-138 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 北川義久: "Isometric deformations of flat tori in the 3-sphere with nonconstant mean curvature"Tohoku Math.J.. 52. 283-298 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 剱持勝衛: "曲面論講義-平均曲率-定曲面入門"培風館. 150 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 前田 定広: "Simpleness and closedness of circles in compact Hermitian symmetric spaces"Tsukuba J. Math.. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 剱持 勝衛: "On minimal surfaces with constant curvature in two-dimensional complex space forms"J. Reine Angew. Math. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 剱持 勝衛: "The classification of the surfaces with parallel mean curvature vector in tow-dimensional complex space forms"Amer. J. Math. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 宮岡 礼子: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces"Tohoku Math. J.. 51. 35-53 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 伊藤 光弘: "A Kummer type construction of self-dual metrics on the connected sum of four complex profective planes"J. Math. Soc. Japan. 52. 139-160 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 佐々木 武: "Globally defined linear connections on FR^1 and S^1"Tohoku Math. J.. 51. 205-212 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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