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Integral geometry and variational problems in homogeneous spaces

Research Project

Project/Area Number 16540051
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

TASAKI Hiroyuki  University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor, 大学院・数理物質科学研究科, 助教授 (30179684)

Co-Investigator(Kenkyū-buntansha) ITOH Mitsuhiro  University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (40015912)
YASUKURA Osami  University of Fukui, Faculty of Engineering, Professor, 工学部, 教授 (00191122)
HASHIMOTO Hideya  Meijo University, Faculty of Science & Technology, Professor, 理工学部, 教授 (60218419)
IKAWA Osamu  Fukushima National College of Technology, Department of General Education, Associate Professor, 一般科, 助教授 (60249745)
KOKUBU Masatoshi  Tokyo Denki University, Department of Mathematical Sciences, Associate Professor, 工学部, 助教授 (50287439)
Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2005: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2004: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordshomogeneous space / integral geometry / differential geometry / Poincare formula / Crofton formula / reflective submanifold / 変分問題
Research Abstract

In the last academic year we established a Crofton formula in Riemannian symmetric spaces by the use of reflective submaifolds, which is totally geodesic. In order to get explicite expression of the Crofton formula we need some geometric invariants of submanifolds. In the case where the head investigator gave an explicit expression of Poincare formula of submanifolds in complex space forms, he introduced a notion of multiple Kahler angle. By the use of the multiple Kahler angle he could obtain an explicit Crofton formula of submanifolds in complex space forms. For the purpose that we extend the class of submanifolds we use in Crofton formula in Riemannian symmetric spaces, we extend the notion of reflective submanifolds to that of weakly reflective submanifolds. Weakly reflective submanifolds are special ones of minimal submanifolds. Some arguments show that austere submanifolds are weakly reflective. In order to get austere submanifolds in the spheres we described a condition for a submanifold to be austere among the orbits of the linear isotropy actions of Riemannian symmetric pairs. As a result of this we obtained a classification of austere orbits in the spheres under the linear isotropy actions. We observed that some of them are invariant under an isometry of the sphere which reverses the submanifold with respect to the normal directions. So we called such submanifolds weakly reflective submanifolds and started our research of weakly reflective submanifolds.

Report

(3 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • Research Products

    (17 results)

All 2006 2005 2004

All Journal Article (17 results)

  • [Journal Article] Geometry of reflective submanifolds in Riemannian symmetric spaces2006

    • Author(s)
      Tasaki
    • Journal Title

      J. Math. Soc. Japan 58・1

      Pages: 275-297

    • NAID

      10017178484

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Geometry of reflective submanifolds in Riemannian symmetric spaces2006

    • Author(s)
      Tasaki
    • Journal Title

      Journal of Math.Soc.Japan 58・1

      Pages: 275-297

    • NAID

      10017178484

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Geometry of reflective submanifolds in Riemannian symmetric spaces2006

    • Author(s)
      Tasaki
    • Journal Title

      J.Math.Soc.Japan 58・1

      Pages: 275-297

    • NAID

      10017178484

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Crofton formulae by reflective submanifolds in Riemannian symmetric spaces2005

    • Author(s)
      Tasaki
    • Journal Title

      Contemporary aspects of complex analysis, differential geometry and mathematical physics

      Pages: 316-325

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Crofton formulae by reflective submanifolds in complex space forms2005

    • Author(s)
      Tasaki
    • Journal Title

      Proceedings of the Ninth International Workshop on Differential Geometry

      Pages: 41-50

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Miyaoka-Yau inequality and complex hyperbolicity2005

    • Author(s)
      Itoh
    • Journal Title

      Topics in almost Hermitian geometry and related fields

      Pages: 105-112

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Variational stability and local rigidity of Einstein metrics2005

    • Author(s)
      Itoh, Nakagawa
    • Journal Title

      Yokohama Math. J. 51・2

      Pages: 103-115

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The modified Yamabe problem and geometry of modified scalar curvatures2005

    • Author(s)
      Itoh
    • Journal Title

      J. Geom. Anal. 15・1

      Pages: 63-81

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Crofton formulae by reflective submanifold in Riemannian symmetric spaces2005

    • Author(s)
      Tasaki
    • Journal Title

      Contemporary aspects of complex analysis, differential geometry and mathematical physics

      Pages: 316-325

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Variational stability and local rigidity of Einstein metrics2005

    • Author(s)
      Itoh, Nakagawa
    • Journal Title

      Yokohama Math.J. 51・2

      Pages: 103-115

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] The modified Yamabe problem and geometry of modified scalar curvatures2005

    • Author(s)
      Itoh
    • Journal Title

      J.Geom.Anal. 15・1

      Pages: 63-81

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Poincare formulas of complex submanifolds2004

    • Author(s)
      Kang, Sakai, Takahashi, Tasaki
    • Journal Title

      Proc.Japan Acad.Ser.A Math.Sci. 80・6

      Pages: 110-112

    • NAID

      120007131208

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Almost Kahler 4-manifolds, L^2-scalar curvature functional and Seiberg-Witten equations2004

    • Author(s)
      Itoh
    • Journal Title

      Internat.J.Math. 15・6

      Pages: 573-580

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Projective geometry of Freudenthal's varieties of certain type2004

    • Author(s)
      Kaji, Yasukura
    • Journal Title

      Michigan Math.J. 52・3

      Pages: 515-542

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Deformations of super-minimal J-holomorphic curves of a 6-dimensional sphere2004

    • Author(s)
      Hashimoto
    • Journal Title

      Tokyo J.Math. 27・2

      Pages: 285-298

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Motion of charged particles in homogeneous Kahler and homogeneous Sasakian manifolds2004

    • Author(s)
      Ikawa
    • Journal Title

      Far East J.Math.Sci. 14・3

      Pages: 283-302

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Flat fronts in hyperbolic 3-space2004

    • Author(s)
      Kokubu, Umehara, Yamada
    • Journal Title

      Pacific J.Math. 216・1

      Pages: 149-175

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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