| Project/Area Number |
12640087
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Sophia University |
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Principal Investigator |
MIYAOKA Reiko Sophia U., Fac. Sc.et Tech., Professor, 理工学部, 教授 (70108182)
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| 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)
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| Project Period (FY) |
2000 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| 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)
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| Keywords | Gauss 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 |
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| 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.
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