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Studies on some open problems concerning flat tori in odd dimensional spheres

Research Project

Project/Area Number 24540066
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionUtsunomiya University

Principal Investigator

KITAGAWA Yoshihisa  宇都宮大学, 教育学部, 教授 (20144917)

Co-Investigator(Renkei-kenkyūsha) AIHARA Yoshihiro  福島大学, 人間発達文化学類, 教授 (60175718)
UMEHARA Masaaki  東京工業大学, 大学院情報理工学研究科, 教授 (90193945)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords微分幾何 / 部分多様体 / 平坦トーラス / 3次元球面 / 直径予想 / 剛性定理 / 正則閉曲線 / 2重接触 / 微分幾何学 / 直径 / 剛性
Outline of Final Research Achievements

Diameter conjecture on flat tori in the unit 3-sphere states that the extrinsic diameter of isometrically immersed flat tori in the unit 3-sphere is equal to π. In 2011, we obtained a theorem which states that the conjecture is true under the assumption that the mean curvature of the immersion is nonnegative or nonpositive. Unfortunately, there was a gap in the proof of a proposition which was a key assertion to prove the theorem. In this research, using a new lemma on the shape of simple loops in the unit 2-sphere, we corrected the proof of the proposition. As a result, we established the theorem.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (5 results)

All 2015 2014 2012

All Journal Article (2 results) (of which Peer Reviewed: 1 results) Presentation (3 results) (of which Invited: 3 results)

  • [Journal Article] Erratum to: Extrinsic diameter of immersed flat tori in S^32014

    • Author(s)
      Y.Kitagawa, M.Umehara
    • Journal Title

      Geometriae Dedicata

      Volume: 171 Issue: 1 Pages: 407-412

    • DOI

      10.1007/s10711-013-9900-z

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 3次元球面内の平坦トーラスに関する直径予想2012

    • Author(s)
      北川義久
    • Journal Title

      数理解析研究所講究録

      Volume: 1817

    • Related Report
      2012 Research-status Report
  • [Presentation] 3次元球面内の平坦トーラスに関する未解決問題2015

    • Author(s)
      北川義久
    • Organizer
      数理科学小研究集会
    • Place of Presentation
      福島大学国際交流会館
    • Year and Date
      2015-01-31
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] S^3 内の平坦トーラスに関する未解決問題2014

    • Author(s)
      北川義久
    • Organizer
      研究集会「部分多様体論・湯沢2014」
    • Place of Presentation
      越後湯沢町・湯沢グランドホテル
    • Year and Date
      2014-11-21
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 3次元球面内の平坦トーラスに関する直径予想2012

    • Author(s)
      北川義久
    • Organizer
      RIMS研究集会「部分多様体と四元数構造」
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Research-status Report
    • Invited

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Published: 2013-05-31   Modified: 2019-07-29  

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