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Submanifolds of homogeneous spaces and Grassmann geometry

Research Project

Project/Area Number 10640066
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo University of Agriculture and Technology

Principal Investigator

MASHIMO Katsuya  Tokyo University of Agriculture and Technology, Faculty of Technology, Professor, 工学部, 教授 (50157187)

Co-Investigator(Kenkyū-buntansha) KODA Takashi  Toyama University, Faculty of Science, Associate Professor., 理学部, 助教授 (40215273)
HASHIMOTO Hideya  Nippon Institute of Technology, Faculty of Technology, Associate Professor., 工学部, 助教授 (60218419)
TASAKI Hiroyuki  University of Tsukuba, Department of Mathematics, Associate Professor., 数学系, 助教授 (30179684)
TOJO Koji  Chiba Institute of Technology, Lecturer., 工学部, 講師 (30296313)
IKAWA Osamu  Fukushima National College of Technology, Department of General Education, Associate Professor., 助教授 (60249745)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2000: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywords6-dimensional sphere / Grassmann Geometry / G_2 / CR部分多様体 / 全実部分多様体 / コンパクトリー群 / カルタン埋め込み / 安定性
Research Abstract

The exceptional simple Lie group G_2 acts naturally on the 6-dimensional sphere S^6.Consider the decomposition of the Grassmann bundle G_p(TS^6) of all p-dimensional subspaces of tangent space of S^6. For a G_2-orbit ν of G_p(TS^6), a submanifold N of S^6 is said to be a ν-submanifold if all of the tangent space of N is contained in ν. We investigated the properties of ν submanifolds.
1. Construction and existence :
(1) Case p=2, there exists a ν-submanifold for any G_2-orbit ν of G_p(TS^6).
(2) Case p=3, the orbit space of G_p(TS^6) is identified with the real projective plane. If a compact ν-submanifold exists ν is contained in a line of the real projective plane.
We studied if the tubes over a J-holomorphic curve in the direction of the first (or second) Normal bundle is a ν-submanifold.
(3) Case p=4, we constructed many 4-dimensional CR submanifolds. But for another orbit ν the existence of ν-submanifold is open.
2. G_2 rigidity of CR submanifols
We gave a condition that two CR submanifolds are G_2 congruent and as its application we gave a characterization of CR submanifolds given by K.Sekigawa.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report

Research Products

(15 results)

All Other

All Publications

  • [Publications] H.Hashimoto and K.Mashimo: "On some 3-dimensional CR-submanifolds in S^6"Nagoya Mathematical Journal. 45. 171-185 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Mashimo and K.Tojo,: "Circles in Riemannian symmetric spaces."Kodai Mathematical Journal. 22. 1-14 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Hashimoto,K.Mashimo and K.Sekigawa,: "On 4-dimensional CR-submanifolds of a 6-dimensional sphere"Advanced Studies in Pure Mathematics. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Hashimoto: "J-holomorphic curves of a 6-dimensional sphere"Tokyo Journal of Mathematics. 23. 137-159 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Hashimoto and K.Mashimo: "On some 3-dimensional CR-submanifolds in S^6"Nagoya Math. J.. 5. 171-185 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Hashimoto, K.Mashimo and K.Sekigawa: "On 4-dimensional CR-submanifolds of a 6-dimensional sphere"To appear in Adv. Studies in Pure Math.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Hashimoto: "J-holomorphic curves of a 6-dimensional sphere"Tokyo J.Math.. 23. 137-159 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Mashimo and K.Tojo: "Circles in Riemannian symmetric spaces."Kodai Mathematical Journal. 22. 1-14 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Hashimoto and K.Mashimo: "On some 3-dimensional CR-submanifolds in S^6"Nagoya Mathematical Journal. 45. 171-185 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Mashimo and K.Tojo, : "Circles in Riemannian symmetric spaces."Kodai Mathematical Journal. 22. 1-14 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Hashimoto,K.Mashimo and K.Sekigawa,: "On 4-dimensional CR-submanifolds of a 6-dimensional sphere"Advanced Studies in Pure Mathematics. (To appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Hashimoto: "J-holomorphic curves of a 6-dimensional sphere"Tokyo Journal of Mathematics. 23. 137-159 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Hashimoto and K.Mashimo: "On some 3-dimensional CR-submanifolds in S^6"Nagaya Mathematical Journal. 156. 171-185 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Mashimo and K.Tojo: "Circles in Riemannian symmetric spaces"Kodai Mathematical Journal. 22. 1-14 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 間下克哉: "カルタン埋め込みの安定性について" 数理解析研究所講究録. 1069. 53-62 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-03-31   Modified: 2016-04-21  

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