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Differential systems and submanifolds theory

Research Project

Project/Area Number 12640087
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

MIYAOKA Reiko  Sophia U., Fac. Sc.et Tech., Professor, 理工学部, 教授 (70108182)

Co-Investigator(Kenkyū-buntansha) KATO Masahide  Sophia U., Fac. Sc.et Tech., Associate Professor, 理工学部, 教授 (90062679)
UCHIYAMA Kouichi  Sophia U., Fac. Sc.et Tech., Professor, 理工学部, 教授 (20053689)
KANEYUKI Soji  Sophia U., Fac. Sc.et Tech., Professor Emeritus, 理工学部, 名誉教授 (40022553)
UMEHARA Masaki  Hiroshima U. Fac. Sc., Professor, 大学院・理学研究科, 教授 (90193945)
YOKOYAMA Kazyi  Sophia U., Fac. Sc.et Tech., Associate Professor, 理工学部, 助教授 (10053711)
田丸 博士  上智大学, 理工学部, 助手 (50306982)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsGauss mapping / Lagrangian submanifolds / Austere submanifolds / Isoparametric hypersurfaces / Shape operators / Completely integrable systems / Surfaces of constant mean curvature / Harmonic maps / 主曲率 / 重複度 / 退化 / スペシャルラグランジアン部分多様体
Research Abstract

We gave a new simple proof of the homogeneity of isoparametric hypersurfaces with six principal curvatures with single multiplicity. Concerning this, we investigate submanifolds in the spheres with degenrate Gauss mapping, and constructed many examples satisfying the equality in the Ferus inequality. All focal submanifolds of isoparametric hypersurfaces and all isoparametric hypersurfaces with degenerate Gauss mapping are austere submanifolds. It is known that the canonical embedding of the normal bundle of a (cone of) austere submanifold gives a special Lagrangian submanifold. We have shown that a proper complete austere submanifolds in R^n have the same homotopy type as a CW complex of dimension not larger than half dimension of the submanifold. This is interesting from the view point of volume minimizing normal bundles. As an inverse problem of splitting Gauss maps of surfaces of constant mean curvature in S^3, we get necessary and sufficient conditions to get surfaces of constant mean curvature from a pair of harmonic maps intoS^2.
Kaneyuki studied orbit of generalized conformal transformation groups. Uchiyama investigated singularities of a non-linear ordinary differential eqquations. Yokoyama completed the theory of extended DS diagram. Tamaru classified isotropy orbits of certain symmetric spaces of non-compact type. Aiyama gave many kind of reperesentation formulas for surfaces, especially Langarnian surfaces in C^2.Ishikawa gave a topological classification of developable surfaces of space curves and studied singularities of contact manifolds. Udagawa described harmonic maps from tori into Grassmannian manifolds by using elliptic functions. He also studied circles in Hermitian symmetric spaces. Umehara studied singularities of curves, ends of minimal surfaces and concerning with Gauss map of these, studied an intrinsic properties of surface with constant curvature 1.Kimura constructied many homogeneous examples of circle and sphere bundles over submanifolds.

Report

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

    (79 results)

All Other

All Publications (79 results)

  • [Publications] R.Aiyama: "A global correspondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2"Proc.Amer.Math.Soc.. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Miyaoka: "A simple proof of the homogeneity of : oparametric hypersurfaces with (g, m)=(6,1)"Geometry and Topology of Submanifolds X. 178-199 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] G.Ishikawa: "Submanifolds with degenerate Gauss mappings in the spheres"Josai Mathematical Monograph Differential Geomerty. 3. 125-138 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kaneyuki: "Compactification of parahermitian symmetric spaces and its applications I : Tube-type realizations"Proceedings of the III International Workshop : Lie Theory and Its Applications in Physics, World Scientific. 63-74 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Balachandran: "Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces"Proc.Indian Acad.Sci.(Math.Sci.). 110,No.2. 225-232 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Balachandran: "Existence of solutions of quasilinear integrodifferential equations with nonlocal condition"Tokyo J.Math.. 23,No.1. 203-210 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ikeda: "Remodeling a DS-diagram into one with E-cycle"Tokyo Jour.Math.. 23. 113-135 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Math.J.. 52. 95-105 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Aiyama: "Kenmotsu type representation formula for spacelike surfaces with prescribed mean in the de Sitter 3-space"Tsukuba J.Math.. 24. 189-196 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Aiyama: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Math.J.. 52. 415-429 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"J.Math.Soc.Japan. 52. 877-898 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Aiyama: "Totally real surfaces in the complex 2-space : Steps in Differential Geometry"Proceedings of the Colloquium on Differential Geometry. July. 25-30 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Aiyama: "Lagrangian surfaces in the conplex 2-space"Proceedings of the 5th International Workshop(Kyungpch U). 25-29 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R.Aiyama: "Constant mean curvature surfaces in the hyperbolic 3-space and harmonic maps"Proceedings of the 5th International Workshop(Kyungpch U). 31-39 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] G.Ishikawa: "Topological classification of the tangent developables of space curves"Jounal of London Math.Soc.. 62-2. 583-598 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] G.Ishikawa: "Solution surfaces of the Menge-Ampeve equation"Differential Geometry and its Application. 14. 113-124 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Kimura: "Minimal immersions of some circle bundles over holomorphic carves in complex quadric to sphere"Osaka Math.J.. 37. 883-903 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] U-Hang Ki: "Geometry of holomorphic distributions of real hypersurfaces in a complex projective space"Czec.Math.J.. 51. 197-204 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Adachi: "Simplenoss and closendess of circles in compact Hernutian symmetricspaces"Tsukuba J.Math.. 24 No.1. 1-13 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kato: "General existence of minimal surfaces of gonus zero with catenokdal ends and pressrihod flux"Communications in Analysis and Geometry. 8. 83-114 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Umehara: "Metrics of constant curvature 1 with three conical singularities on the 2-sphere"Illinois Journal of Mathematics. 44 No.1. 72-94 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A.I.Bobenko: "Monodromy of isometric deformation of CMC surfaces"Hiroshima Mathematical Journal. 31. 291-297 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 宮岡礼子: "Lie sphere geometry and Lie contact structure"数理解析研究所講究録. 1150. 143-146 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 宮岡礼子: "Elie cartanの業績-S.S.ChernとC.Chevaleyによる解説"数理解析研究所講究録. 1206. 1-31 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Faraut: "Analysis And Geometry On Complex Homogeneous Domains Progress in Math Vol.185"Birkhauser. (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 石川剛郎: "代数曲線と特異点、特異点の数理、第4巻"共立出版. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] R. Aiyama: "A global correspondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2"Proc. Amer. Math. Soc.. 128. 939-941 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Miyaoka: "A simple proof of the homogeneity of isoparametric hypersurfaces with (g, m) = (6, 1)"Geometry and Topology of Submanifolds X. 178-199 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] G. Ishikawa: "Submanifolds with degenerate Gauss mappings in the spheres"Josai Mathematical Monograph. Differential Geometry. 3. 125-138 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S. Kaneyuki: "Compactification of parahermitian symmetric spaces and its applicationsI : Tube-type realizations"Proceedings of the III International Workshop : Lie Theory and Its Applications in Physics, World Scientific. 63-74 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] K. Balachandran: "Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces"Proc. Indian Acad. Sci. (Math. Sci.). 110, No.2. 225-232 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] K. Balachandran: "Existence of solutions of quasilinear integrodifferential equations with nonlocal condition"Tokyo J. Math.. 23.No1. 203-210 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] H. Ikeda: "Remodeling a DS-diagram into one with E-cycle"Tokyo Jour. Math.. 23. 113-135 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Aiyama: "Kenmotsu type respresentation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Math. J.. 52. 95-105 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Aiyama: "Kenmotsu type representation formula for spacelike surfaces with prescribed mean in the de Sitter 3-space"Tsukuba J. Math.. 24. 189-196 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Aiyama: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Math. J.. 52. 415-429 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"J. Math. Soc. Japan. 52. 877-898 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Aiyama: "Totally real surfaces in the complex 2-space : Steps in Differential Geometry"Proceedings of the Colloquium on Differential Geometry. July. 25-30 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Aiyama: "Lagrangian surfaces in the complex 2-space"Proceedings of the 5th International Workshop(Kyungpooh U.). 25-29 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Aiyama: "Constant mean curvature surfaces in the hyperbolic 3-space and harmonic maps"Proceedings of the 5th International Workshop(Kyungpooh U.). 31-39 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] G. Ishikawa: "Topological classification of the tangent developables of space curves"Journal of London Math. Soc.. 62-2. 583-598 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] G. Ishikawa: "Solution surfaces of the Monge-Ampere equation"Differential Geometry and its Applications. 14. 113-124 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Kimura: "Minimal immersions of some circle bundles over holomorphic curves in complex quadric to sphere"Osaka Math. J.. 37. 883-903 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] U-Hang Ki: "Geometry of holomorphic distributions of real hypersurfaces in a complexprojective space"Czec. Math. J.. 51. 197-204 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T. Adachi: "Simpleness and closedness of circles in compact Hermitian symmetricspaces"Tsukuba J. Math.. 24 No1. 1-13 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] S. Kato: "General existence of minimal : surfaces of genus zero with catenoidal ends and prescribed flux"Communications in Analysis and Geometry. 8. 83-114 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Umehara: "Metrics of constant curvature 1 with three conical singularities on the 2-sphere"Illinois Journal of Mathematics. 44 No1. 72-94 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] A. I. Bobenko: "Monodromy of isometric deformation of CMC surfaces"Hiroshima Mathematical Journal. 31. 291-297 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 宮岡 礼子: "Lie sphere geometry and Lie contact structure"数理解析研究所講究録. 1150. 143-146 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 宮岡 礼子: "Elie Cartanの業績-S. S. ChernとC. Chevaleyによる解説"数理解析研究所講究録. 1206. 1-31 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 宮岡 礼子: "等径超曲面今昔-Elie Cartanと21世紀"数理解析研究所講究録. 1206. 32-44 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 宮岡 礼子: "On special Lagragian submanifolds"数理解析研究所講究録. 1236. 1-8 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 宮岡 礼子: "等径超曲面再訪"数学. 53. 18-33 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 梅原 雅顕: "3次元双曲型空間における平坦な曲面"数理解析研究所講究録. 1236. 49-59 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 金行壯二: "Recent Development in Geometry of Parahermitian Symmetric spaces"数理解析研究所講究録. 1150. 11-13 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 相山玲子: "A construction of Lagrangian surfaces in the complex 2-space"数理解析研究所講究録. 1206. 121-133 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 石川剛郎: "Recent Results on Developable Submanifolds : Examples and Classifications"数理解析研究所講究録. 1150. 134-141 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] J. Faraut: "Analysis And Geometry On Complex Homogeneous Domains Progress in Math vol.185"Birkhauser. (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 石川 剛郎: "代数曲線と特異点, 特異点の数理. 第4巻"共立出版. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] R.Aiyama: "Aglohal crrespondence between CMC-surfaces in S^3 and pain of non-conformal harmonic mops in liS^2"Proc.Amer.Math.Soc.. 128. 939-941 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Miyaoka: "A simple proct of the homogeneity of isoparametric hyper surfaces with (g.m)=(6.1)"Geometry and Topology of Submamifolds. X. 178-199 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Miyaoka: "Lie spheare geometry and Lie contact structures"数理解析研究所講究録. 1150. 143-146 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 宮岡礼子: "等径超曲面再訪"数学. 53. 18-33 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Goo Ishikawa: "Topological classification of the tangent developables of a nace curres"Jour.of London Math.Soc.. 62. 583-598 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Ikeda: "Remodelings DS-diagram into one with E-cycle Tokyo Jour.Hath."

    • Related Report
      2000 Annual Research Report
  • [Publications] Goo Ishikawa : "Solution surfaces of monge-Ampire equation"Diff.Geom.and Appl.. (to appear). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Aiyama: "Kenmoteu tyre represeritations for mula for surfaces with prescribed mean curvature in the 3-sphere"Tohokci Math.J.. 52. 95-105 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Aiyama: "Kenmoten tyre represeritation for mula for space like surfaces with prescribed mean curvature in the de Sittu 3-space"Tsukuha J.Math. 24. 189-196 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Aiyama: "Minimal mops Between hyperbolic dish and generalized Gaussmans of maximal surfaces in the anti de sitter 3-spaces"Tohoka J,Math. 52. 415-429 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Aiyama: "Kenmotsu tyre represeritation for mula for surfaces with prescribed mean curvature -in the hyperbolic 3-space"J.Math.Soc.Japan. 52. 877-898 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kaneyuki: "Comfactification of parahermitian symmetri spaces and its application"Lie Theory and its Application in Physics. 63-74 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kato: "General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux"Comm.Analy and Geom.. 8. 83-114 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Umehare: "Metrics of constand curvature 1 with three conical singularities on the 2-sphere"Fllinois J.of Math.. 44. 72-94 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Kumura: "Minimal immesscions of some cricle bandle over holomorphic curves in complex quadric to sphere"Osaha.Math.J.. 37. 883-903 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Adachi: "Simpleness and closedness of circles in compact Hermitian symmetric spaces"Tsukuha J.Math.. 24. 1-13 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Balachandran: "Existence of solutions of nonlinear integro differential equations of Sobolev type with nonlocal condition in Banach spaces "Proc.Indian Acad.Sci.. 110. 225-232 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Balanchamdran : "Existence of solutions of quanilinear integrodifferential equations with nonlocal condition"Tokyo J.Math. 23. 203-210 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] G.Ishikura: "Submanifolds with degenerate Gauss mapping"Josai Mathematical monograph 3,Differential Geometry. 3. 125-138 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] I.Faraut: "Analysis and Geometry on Complex Homogeneous Domains."Progress in Math.,Birkhanser.. 540 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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