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2007 Fiscal Year Final Research Report Summary

Research of submanifolds in a symmetric space by using infinite dimensional geometry

Research Project

Project/Area Number 18540099
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo University of Science

Principal Investigator

KOIKE Naoyuki  Tokyo University of Science, Faculty of Science, Associate Professor (00281410)

Co-Investigator(Kenkyū-buntansha) YOSHIOKA Akira  Tokyo University of Science, Faculty of Science, Professor (40200935)
TAMARU Hiroshi  Hiroshima University, Faculty of Science, Associate Professor (50306982)
SAKAI Takashi  Osaka City University, Faculty of Science, Special Duty Assistant (30381445)
Project Period (FY) 2006 – 2007
KeywordsDifferential Geoimetry
Research Abstract

As the first study result of 2006, we obtained a Chevalley type restriction theorem for a proper complex equifocal submanifold in a symmetric space of non-compact type. The proof was performed by investigating the lift of its complexification to some infinite dimensional anti-Kaehlerian space. Here we note that principal orbits of Hermann type actions on the symmetric Name are proper complex equifocal submanifolds. This research was performed by the head investigator. As the second study result, we almost classified complex hyperpolar actions with total geodesic orbit on a symmetric space of non-compact type. Here we note that principal orbits of complex hyperpolar actions are complex equifocal submanifolds and that conversely homogeneous complex equifocal submanifolds war as principal orbits of complex hyperpolar actions. Also, we note that Hermann type actions are complex hyperpolar actions. This research was performed by the head investigator. As the third study result, we classified cohomogeneity one actions on rankone symmetric spaces of non-compact type. This research was performed by Professor Hiroshi Tamaru of the investigator and Professor Jurgen Berndt.
As the first study result of 2007, we completed almost the proof of the homogeneity theorem for irreducible proper complex equifocal submanifolds of codimension greater than one in a symmetric space of non-compact type. The proof was performed by investigating the lift of its complexification to some infinite dimensional anti-Kaehlerian space. This research was performed by the head investigator. As the second study result, we completed almost the proof of the non-existence theorem of equifocal submanifolds with non-Bat section in an irreducible symmetric space of rank greater than one, which is an open problem. This research was performed by the head investigator.

  • Research Products

    (68 results)

All 2008 2007 2006 Other

All Journal Article (22 results) (of which Peer Reviewed: 11 results) Presentation (44 results) Book (2 results)

  • [Journal Article] Noncompact homogeneous Einstein manifolds attached to graded Lie algebras2008

    • Author(s)
      田丸 博士
    • Journal Title

      Mathematische Zeitschrift (未定)(掲載決定済)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Lie groups locally isomorphic to generalized Heisenberg groups2008

    • Author(s)
      田丸 博士
    • Journal Title

      Proceedings of the American Mathematical Society (未定)(掲載決定済)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] A Chevalley type restriction theorem for a proper complex equifocal submanifold2007

    • Author(s)
      小池 直之
    • Journal Title

      Kodai Mathematical Journal 30

      Pages: 280-296

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Complex hyperpolar actions with a totally geodesic orbit2007

    • Author(s)
      小池 直之
    • Journal Title

      Osaka Journal of Mathematics 44

      Pages: 491-503

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Geometric objects in an approach to quantum geometry2007

    • Author(s)
      吉岡 朗
    • Journal Title

      Progress in Mathematics 252

      Pages: 303-324

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Non-formal deformation quantization of Frechet-Poisson algebras:The Heisenberg Lie algebra case2007

    • Author(s)
      吉岡 朗
    • Journal Title

      Contemporary Mathematics 434

      Pages: 99-123

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Orderings and non-formal deformation quantization2007

    • Author(s)
      吉岡 朗
    • Journal Title

      Letters in Mathematical Physics 82

      Pages: 153-175

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Cohomogeneity one actions on noncompact symmetric spaces of rank one2007

    • Author(s)
      田丸 博士
    • Journal Title

      Transactions of the American Mathematical Society 359

      Pages: 3425-3438

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Transferred kinematic formulae in two point homogeneous spaces2007

    • Author(s)
      酒井 高司
    • Journal Title

      Tsukuba Journal of Mathematics 31

      Pages: 343-354

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] A Chevalley type restriction theorem for a proper complex equifocal submanifold2007

    • Author(s)
      Naoyuki, Koike
    • Journal Title

      Kodai, Mathematical Journal 30

      Pages: 280-296

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Complex hyperpolar actions with a totally geodesic orbit2007

    • Author(s)
      Naoyuki, Koike
    • Journal Title

      Osaka Journal of Mathematics 44

      Pages: 491-503

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Geometric objects in an approach to quantum geometry2007

    • Author(s)
      Akira, Yoshioka
    • Journal Title

      Progress in Mathematics 252

      Pages: 303-324

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Non-formal deformation quantization of Frechet-Poisson algebras : The Heisenberg Lie algebra case2007

    • Author(s)
      Akira, Yoshioka
    • Journal Title

      Contemporary Mathematics 434

      Pages: 99-123

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Orderings and non-formal deformation quantization2007

    • Author(s)
      Akira, Yoshioka
    • Journal Title

      Letters in Mathematical Physics 82

      Pages: 153-175

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Cohomogeneity one actions on noncompact symmetric spaces of rank one2007

    • Author(s)
      Hiroshi, Tamaru
    • Journal Title

      Transactions of the American Mathematical Society 359

      Pages: 3425-3438

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Transferred kinematic formulae in two point homogeneous spaces2007

    • Author(s)
      Takashi, Sakai
    • Journal Title

      Tsukuba Journal of Mathematics 31

      Pages: 343-354

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A splitting theorem for proper complex equifocal submanifolds2006

    • Author(s)
      小池 直之
    • Journal Title

      Tohoku Mathematical Journal 58

      Pages: 393-417

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] The variational formulas of the volume function in the equiaffine geometry2006

    • Author(s)
      小池 直之
    • Journal Title

      Hokkaido Mathematical Journal 35

      Pages: 719-751

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] A splitting theorem for proper complex equifocal submanifolds2006

    • Author(s)
      Naoyuki, Koike
    • Journal Title

      Tohoku Mathematical Journal 58

      Pages: 393-417

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The variational formulas of the volume function in the equiaffine geometry2006

    • Author(s)
      Naoyuki, Koike
    • Journal Title

      Hokkaido Mathematical Journal 35

      Pages: 719-751

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Noncompact homogeneous Einstein manifolds attached to graded Lie algebras

    • Author(s)
      Hiroshi, Tainaru
    • Journal Title

      Mathematische Zeitschrift (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Lie groups locally isomorphic to generalized Heisenberg groups

    • Author(s)
      Hiroshi, Tamaru
    • Journal Title

      Proceedings of the American Mathematical Society (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Non-existence theorem of equifocal submanifolds with non-flat section2008

    • Author(s)
      小池 直之
    • Organizer
      日本数学会2008年度会
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-23
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] The complexification of a Riemannian symmethric space and Hermann type action2008

    • Author(s)
      小池 直之
    • Organizer
      日本数学会2008年度会
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-23
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Non-existence theorem of equifocal submnaifolds with non-flat section2008

    • Author(s)
      Naoyuki, Koike
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Kinki University
    • Year and Date
      2008-03-23
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] The complexification of a Riemannian symmetric space and Hermann type action2008

    • Author(s)
      Naoyuki, Koike
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Kinki University
    • Year and Date
      2008-03-23
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Parabolic subgroups of semisimple Lie groups and noncompact homogeneous Einstein manifolds2008

    • Author(s)
      田丸 博士
    • Organizer
      第3回日中友好幾何学研究集会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2008-01-29
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Parabolic subgroups of semisimple Lie groups and noncompact homogeneous Einstein manifolds2008

    • Author(s)
      Hiroshi, Tamaru
    • Organizer
      The third Japan-China friendship conference of Geometry
    • Place of Presentation
      Nagoya University
    • Year and Date
      2008-01-29
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Weakly reflective orbits and tangentially degenerate orbits of s-representations I,II2007

    • Author(s)
      酒井 高司
    • Organizer
      Mini International Workshop on Lagrangian Submanifolds and Related Fields
    • Place of Presentation
      Kyungpook National University
    • Year and Date
      20071206-08
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Weakly reflective orbits and tangentially degenerate orbits of s-representations I, II2007

    • Author(s)
      Takashi, Sakai
    • Place of Presentation
      Kyungpook National University
    • Year and Date
      20071206-08
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 複素双曲空間内の等質超曲面の分類2007

    • Author(s)
      田丸 博士
    • Organizer
      部分多様体論・湯沢2007
    • Place of Presentation
      湯沢グランドホテル
    • Year and Date
      20071122-23
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] The classification of homogeneous hypersurfaces in the complex hyperbolic space2007

    • Author(s)
      Hiroshi, Tamaru
    • Organizer
      Submanifold theory-Yuzawa 2007
    • Place of Presentation
      Yuzawa Grand Hotel
    • Year and Date
      20071122-23
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds2007

    • Author(s)
      田丸 博士
    • Organizer
      大阪市立大学数学研究所微分幾何学ワークショップ「部分多様体の幾何学における有限次元および無限次元リー理論的方法」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2007-10-04
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds2007

    • Author(s)
      Hiroshi, Tamaru
    • Organizer
      Workshop in finite dimensional and infinite dimensional Lie theoretic methods in submanifold geometry
    • Place of Presentation
      Osaka City University
    • Year and Date
      2007-10-04
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Weakly reflective orbits and tangentially degenerate orbits of s-representations2007

    • Author(s)
      酒井 高司
    • Organizer
      大阪市立大学数学研究所微分幾何学ワークショップ「部分多様体の幾何学における有限次元および無限次元リー理論的方法」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2007-10-03
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Weakly reflective orbits and tangentially degenerate orbits of s-representations2007

    • Author(s)
      Takashi, Sakai
    • Organizer
      Workshop in finite dimensional and infinite dimensional Lie theoretic methods in submanifold geometry
    • Place of Presentation
      Osaka City University
    • Year and Date
      2007-10-03
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Isoparametric submanifolds in a Riemannian symmetric space of non-compact type and infinite dimensional anti-Kaehlerian geometry2007

    • Author(s)
      小池 直之
    • Organizer
      大阪市立大学数学研究所微分幾何学ワークショップ「部分多様体の幾何学における有限次元および無限次元リー理論的方法」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2007-10-01
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Isoparametric submanifolds in a Riemannian symmetric space of non-compact type and infinite dimensional anti-Kaehlerian geometry2007

    • Author(s)
      Naoyuki, Koike
    • Organizer
      Workshop in finite dimensional and infinite dimensional Lie theoretic methods in submanifold geometry
    • Place of Presentation
      Osaka City University
    • Year and Date
      2007-10-01
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] On non-existenceness of equifocal submanifolds with non-flat section2007

    • Author(s)
      小池 直之
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On curvature-adapted and proper complex equifocal submanifolds2007

    • Author(s)
      小池 直之
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Gauss写像が退化する軌道と弱鏡映軌道2007

    • Author(s)
      酒井 高司
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On non-existenceness of equifocal submnaifolds with non-flat section2007

    • Author(s)
      Naoyruki, Koike
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] On curvature-adapted and proper complex equifocal submanifolds2007

    • Author(s)
      Naoyuki, Koike
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Orbits with degenerate Gauss map and weakly reflective Orbits2007

    • Author(s)
      Takashi, Sakai
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds2007

    • Author(s)
      田丸 博士
    • Organizer
      第54回幾何学シンポジウム
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2007-08-26
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Gauss写像が退化する軌道と弱鏡映軌道2007

    • Author(s)
      酒井 高司
    • Organizer
      第54回幾何学シンポジウム
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2007-08-26
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds2007

    • Author(s)
      Hiroshi, Tamaru
    • Organizer
      The 54-th Symposium of Geometry
    • Place of Presentation
      Kagoshima University
    • Year and Date
      2007-08-26
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Orbits with degenerate Gauss map and weakly reflective Orbits2007

    • Author(s)
      Takashi, Sakai
    • Organizer
      The 54-th Symposium of Geometry
    • Place of Presentation
      Kagoshima University
    • Year and Date
      2007-08-26
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Ordering problem in star product and star product identities2007

    • Author(s)
      吉岡 朗
    • Organizer
      XXVI-th Workshop in geometric methods in physics
    • Place of Presentation
      Bialowieza,Poland
    • Year and Date
      2007-07-07
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Ordering problem in star product and star product Identities2007

    • Author(s)
      Akira, Yoshioka
    • Organizer
      XXIV-th Workshop in geometric methods in Physics
    • Place of Presentation
      Bialowieza, Poland
    • Year and Date
      2007-07-07
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] On the geometry of orbits of s-representations2007

    • Author(s)
      酒井 高司
    • Organizer
      京都大学数理解析研究所研究集会「変換群の理論とその応用」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-05-31
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On the geometry of orbits of s-representations2007

    • Author(s)
      Takashi, Sakai
    • Organizer
      Theory of transformation group and its applications
    • Place of Presentation
      Kyoto University Mathemati-cal principle analysis laboratory
    • Year and Date
      2007-05-31
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 弱鏡映軌道とaustere軌道の分類2007

    • Author(s)
      酒井 高司
    • Organizer
      日本数学会2007年度会
    • Place of Presentation
      埼玉大学
    • Year and Date
      2007-03-29
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] The classification of weakly reflective orbits and austere orbits2007

    • Author(s)
      Takashi, Sakai
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Saitama University
    • Year and Date
      2007-03-29
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 擬リーマン多様体間の写像の複素化とアンチケーラー幾何2007

    • Author(s)
      小池 直之
    • Organizer
      日本数学会2007年度会
    • Place of Presentation
      埼玉大学
    • Year and Date
      2007-03-27
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] プロパー複素等焦部分多様体の完備な複素化の構成法とその構造2007

    • Author(s)
      小池 直之
    • Organizer
      日本数学会2007年度会
    • Place of Presentation
      埼玉大学
    • Year and Date
      2007-03-27
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] The complexification of a map between pseudo-Riemanninan manifolds and anti-Kaehlerian geometry2007

    • Author(s)
      Naoyuki, Koike
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Saitama University
    • Year and Date
      2007-03-27
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] A construction of the complete complexification of a proper complex equifocal submanifold and its structure2007

    • Author(s)
      Naoyuki, Koike
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Saitama University
    • Year and Date
      2007-03-27
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Cohomogeneity one actions on symmetric spaces of rank one,and of higher rank2006

    • Author(s)
      田丸 博士
    • Organizer
      Ireland-Japan Workshop on Geometry and Dynamical Systems
    • Place of Presentation
      慶応大学
    • Year and Date
      2006-12-18
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Cohomogeneity one actions on symmetric spaces of rank one and of higher rank2006

    • Author(s)
      Hiroshi, Tamaru
    • Organizer
      Ireland-Japan Workshop on Geometry and, Dynamical Systems
    • Place of Presentation
      Keio University
    • Year and Date
      2006-12-18
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 弱鏡映部分像多様体とaustere部分多様体2006

    • Author(s)
      酒井 高司
    • Organizer
      部分多様体論・湯沢2006
    • Place of Presentation
      湯沢グランドホテル
    • Year and Date
      2006-12-01
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Weakly reflective submanifolds and austere submanifolds2006

    • Author(s)
      Takashi, Sakai
    • Organizer
      Submanifold theory-Yuzawa 2006
    • Place of Presentation
      Yuzawa Grand Hotel
    • Year and Date
      2006-12-01
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 非コンパクト等質アインシュタイン多様体について2006

    • Author(s)
      田丸 博士
    • Organizer
      幾何学阿蘇研究集会
    • Place of Presentation
      休暇村南阿蘇
    • Year and Date
      2006-09-13
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] 弱鏡映部分像多様体とaustere部分多様体2006

    • Author(s)
      酒井 高司
    • Organizer
      幾何学阿蘇研究集会
    • Place of Presentation
      休暇村南阿蘇
    • Year and Date
      2006-09-13
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On noncompact homogeneous Einstein manifolds2006

    • Author(s)
      Hiroshi, Tamaru
    • Organizer
      Aso research meeting of geometry
    • Place of Presentation
      Minamiaso vacation village
    • Year and Date
      2006-09-13
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Weakly reflective submanifolds and austere submanifolds2006

    • Author(s)
      Takashi, Sakai
    • Organizer
      Aso research meeting of geometry
    • Place of Presentation
      Minamiaso vacation village
    • Year and Date
      2006-09-13
    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] From Geometry to Quantum Mechanics, in Honor of Hdeki Omori(Progress in Mathematics voL 252)2007

    • Author(s)
      吉岡 明
    • Total Pages
      324
    • Publisher
      Birkhauser
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] From Geometry of Quantum Mechanics in Honor of Hideki Omori (Progress in Mathematics vol.252)2007

    • Author(s)
      Akira, Yoshioka
    • Total Pages
      324
    • Publisher
      Birkhauser
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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