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Research on the structure of ideal boundaries of piecewise Riemannian polyhedra from the viewpoint of total curvature

Research Project

Project/Area Number 13640060
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionIbaraki University

Principal Investigator

OHTSUKA Fumiko  Ibaraki University, Faculty of Science, Associate Professor, 理学部, 助教授 (90194208)

Co-Investigator(Kenkyū-buntansha) ITOH Jin-ichi  University of Kumamoto, Faculty of Education, Professor, 教育学部, 教授 (20193493)
OOSHIMA Hideaki  Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (70047372)
URABE Tohsuke  Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (70145655)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Keywordspolyhedron / total curvature / constant curvature / 平坦な多面体
Research Abstract

The field of our study is piecewise Riemannian 2-polyhedra, which is a combinatorial 2-polyhedra. such that each 2-simplex is isometric to a triangle bounded by three smooth curves on some Riemannian 2-manifold. The purpose in this research project is to investigate some properties of noncompact piecewise Riemannian 2-polyhedra from the viewpoint of total curvature.
Main results in this project is the following.
1."Total excess on length surfaces"by Y.Machigashira and F.Ohtsuka : On length spaces satisfying some suitable conditions, which may contain some discrete singular points, we generalize the notion of total curvature (we call it total excess in this paper) and prove some generalized theorems.
2."Total curvature of noncompact piecewise Riemannian 2-polyhedra"by J.Itoh and F.Ohtsuka : We introduce two kind of definitions of total curvature (total curvature and w-total curvature) on piecewise Riemannian 2-polyhedra and obtain some generalized theorems for them admitting total curvature and observe the difference of total curvature and w-total curvature.
3."Structure of flat piecewise Riemannian 2-polyhedra"by F.Ohtsuka : We give a definition of flatness of piecewise Riemannian 2-polyhedra and characterize them.
Henceforth I would like to investigate the structure of 2-polyhedra of constant curvature and the relationship between polyhedra and its ideal boundaries and so on.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (33 results)

All 2005 2004 2001 Other

All Journal Article (16 results) Publications (17 results)

  • [Journal Article] On the length of the cut locus for finitely many points2005

    • Author(s)
      J.Itoh, T.Zamfirescu
    • Journal Title

      Advanced Geometry 5(発表予定)

      Pages: 97-106

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Farthest points and cut loci on some degenerate convex surface2004

    • Author(s)
      J.Itoh, C.Vilcu
    • Journal Title

      J.of Geometry 80

      Pages: 106-120

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Structure of flat piecewise Riemannian 2-polyhedra2004

    • Author(s)
      F.Ohtsuka
    • Journal Title

      Math.J.Ibaraki Univ. 36

      Pages: 57-64

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Farthest points and cut loci on some degenerate convex surfaces2004

    • Author(s)
      J.Itoh, C.Vilcu
    • Journal Title

      J.of Geometry 80

      Pages: 106-120

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Structure of flat piecewise Riemannian 2-polyhedra2004

    • Author(s)
      Fumiko Ohtsuka
    • Journal Title

      Math.J.Ibaraki Univ. 36

      Pages: 57-64

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The total absolute curvature of nonclosed curves in S^22004

    • Author(s)
      K.Enomoto, J.Itoh
    • Journal Title

      Results in Math. 45

      Pages: 21-34

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The total absolute curvature of nonclosed curves in S^2 II2004

    • Author(s)
      K.Enomoto, J.Itoh
    • Journal Title

      Results in Math. 45

      Pages: 230-240

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Total excess on length surfaces2001

    • Author(s)
      Y.Machigashira, F.Ohtsuka
    • Journal Title

      Math.Ann. 319

      Pages: 675-706

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Lipschitz continuity of the distance function to the cut locus2001

    • Author(s)
      J.Itoh, M.Tanaka
    • Journal Title

      Trans.of AMS 353

      Pages: 21-40

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A Sard theorem for the distance function2001

    • Author(s)
      J.Itoh, M.Tanaka
    • Journal Title

      Math.Ann. 320

      Pages: 1-10

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Total excess on length surfaces2001

    • Author(s)
      Yoshiroh Machigashira, Fumiko Ohtsuka
    • Journal Title

      Math.Ann. 319

      Pages: 675-706

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Lipschitz continuity of the distance function to the cut locus2001

    • Author(s)
      Jin-ichi Itoh, Minoru Tanaka
    • Journal Title

      Trans.Amer.Math.Soc. 353

      Pages: 21-40

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A Sard theorem for the distance function2001

    • Author(s)
      Jin-ichi Itoh, Minoru Tanaka
    • Journal Title

      Math.Ann. 320

      Pages: 1-10

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Total curvature of noncompact piecewise Riemannian 2-polyhedra

    • Author(s)
      J.Itoh, F.Ohtsuka
    • Journal Title

      Tsukuba J.Math. (accepted)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Total curvature of noncompact piecewise Riemannian 2-polyhedra

    • Author(s)
      Jin-ichi Itoh, Fumiko Ohtsuka
    • Journal Title

      Tsukuba J.Math. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Total curvature of noncompact piecewise Riemannian 2-polyhedra

    • Author(s)
      J.Itoh, F.Ohtsuka
    • Journal Title

      Tsukuba J.Math. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] H.Oshima: "The group of self homotopy classes of SO(4)"J.Pure Appl.Algebra. 185. 193-205 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Itoh, T.Zamfirescu: "Acute triangulations of the regular icosahedral surface"Discrete and Computational Geometry. (掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Itoh, T.Zamfirescu: "On the length of the cut locus for finitely many points"Advanced Geometry. (掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Enomoto, J.Itoh: "The total absolute curvature of nonclosed curves in S^2"Results in Math.. (掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Enomoto, J.Itoh: "The total absolute curvature of nonclosed curves in S^2 II"Results in Math.. (掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] Fumiko Ohtsuka: "Structures of Flat Piecewise Riemannian 2-Polyhedra"Abstracts of Short Communications and Poster Sessions, International Congress of Mathematicians, Baijing 2002. 71 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Arkowitz, J.Strom, H.Oshima: "Noncommutativity of the group of self homotopy classes of classical simple Lie groups"Topology Appl.. 125. 87-96 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Arkowitz, J.Strom, H.Oshima: "The inverses of an H-space"Manuscripta Math.. 106. 399-408 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Itoh, T.Zamfirescu: "Acute triangulations of the regular icosahedral surface"Discrete and Computational Geometry. (accepted).

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Itoh, T.Zamfirescu: "On the length of the cut locus on surfaces"Rendiconti del Circolo Matematico di Palermo Serie II. 70. 53-58 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Itoh, T.Zamfirescu: "Acute triangulations of triangles on the sphere"Rendiconti del Circolo Matematico di Palermo Serie II. 70. 59-64 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshiroh Machigashira, Fumiko Ohtsuka: "Total excess on length surfaces"Math. Ann.. 319. 675-706 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hideaki Ooshima, Nobuaki Yagita: "Non commutativity of self homotopy groups"Kodai Math. J.. 24. 15-25 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hideaki Ooshima: "Multiplications on a localized Hopf space"Math. J. Ibaraki Univ.. 33. 17-21 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Jin-ichi Itoh, Minoru Tanaka: "The Lipschitz continuity of the distance function to the cutlocus"Trans. of AMS. 353. 21-40 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Jin-ichi Itoh, Minoru Tanaka: "A Sard theorem for the distance function"Math. Ann.. 320. 1-10 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Jin-ichi Itoh: "Essential cutlocus of on a surface"Tohoku Math. Publ. 20. 53-59 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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