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Geometry of spaces and groups with negative eurvature

Research Project

Project/Area Number 08454020
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

YAMAGUCHI Takao  Kyushu University, Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (00182444)

Co-Investigator(Kenkyū-buntansha) NISHI Haruko  Kyushu University, Graduate School of Mathematics, Research Associate, 大学院・数理学研究科, 助手 (90274430)
YOKOTA Yoshiyuki  Kyushu University, Graduate School of Mathematics, Assistant Professor, 大学院・数理学研究科, 講師 (40240197)
KAMATA Masayoshi  Graduate School of Mathematics, Kyushu University, Professor, 大学院・数理学研究科, 教授 (60038495)
KATO Mitsuyoshi  Graduate School of Mathematics, Kyushu University, Professor, 大学院・数理学研究科, 教授 (60012481)
SHIOHAMA Katsuhiro  Graduate School of Mathematics, Kyushu University, Professor, 大学院・数理学研究科, 教授 (20016059)
Project Period (FY) 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 1996: ¥3,900,000 (Direct Cost: ¥3,900,000)
KeywordsAlexandrov spaces / simplicial volume / isometry group / convergence theory / collapsing
Research Abstract

We have mainly studied the simplicial volumes, the isometry groups, the finiteness property for singular spaces with negative curvature, and established the theory of collapsing for 3-manifolds under a lower curvature bound. (1) We have shown that for a generalized psuedomanifold the mass of its fundamental homology class coincides with the total volume, and obtained a relation between the simplicial volume, the curvature bound and the volume. We have also obtained the generalization of Gromov-Thurston inequality for spaces with singularities of codimension one. (2) We have observed the isometry group of an Alexandrov space with curvature bounded above, and proved that the subgroup leaving in variant a compact set is a Lie group. By using this result basically, we have found out some properties of the isometry group similar to those of trees. As an application, we have shown that the isometry group of a simply connected space with nonpositive curvature is a Lie group if the ideal boundary is compact with respect to the Titz metric. (3) In the joint work with Takashi Shioya, we study the collapsing phenomena of three-manifolds under a lower curvature bound. More precisely, by finding out some methods about critical points of distance functions, we have succeeded to describe collapsed manifolds in terms of the limit space. (4) We have shown that the finiteness of homotopy types of spaces with curvature <less than or equal> -1 and with bounds on dimameter and volume. Now we are trying to extend this result to the finiteness of homeomorphisms under an addition hypothesis on the volume of each strata of the dimension stratific

Report

(2 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • Research Products

    (15 results)

All Other

All Publications (15 results)

  • [Publications] Takao Yamaguchi: "Development of convergence theory for Riemannian manifolds." Sugaku. 47. 46-61 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Katsuhiro Shiohama: "Cut loci and distance spheres on Alexandrov surfaces" Seminair es at Congres,Collection SMF. No.1. 531-560 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Masayoshi Kamata: "On multiple points of a self-transverse immersion" Kyushu Journal of Mathematics. 50.2. 275-283 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Yoshiyuki Yokota: "Topological invariants of graphs in 3-space" Topology. 35. 77-87 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Haruko Nishi: "Pentagonal Structure of the configuration space of five points in the real proiective line II" (in preparation).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Takao Yamaguchi: "Development of convergence theory for Riemannian manifolds" Sugaku. No.47. 46-61 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Katsuhiro Shiohama: "Cut loci and distance Spheres on Alexandrov Surfaces" Seminair es et Congres, Collection SMF. No.1. 531-560 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Masayoshi Kamata: "On multiple points of a self-transverse immersion." Kyushu Journal of Mathematics. No.50 vol.2. 275-287 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Yoshiyuki Yokota: "Topological invariants of graph in 3-space" Topology. No.35. 77-87 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Haruko Nishi: "Pentagonal Structure of the configuration space of five points in the real projective line II" (in preparation).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Takao Yamaguchi: "Development of convergence theory for Riemannian manifolds." Sugaku. 47. 46-61 (1995)

    • Related Report
      1996 Annual Research Report
  • [Publications] Katsuhiro Shiohama: "Cut loci and distance spheres on Alexandrov surfaces" Seminair es et Congres,Collection SMF. No.1. 531-560 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Masayoshi Kamata: "On multiple points of a self-transverse immersion" Kyusyu Journal of Mathematics. 50.2. 275-283 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Yoshiyuki Yokota: "Topological invariants of graphs in 3-space" Topology. 35. 77-87 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Haruko Nishi: "Pentagonal Structure of the configuration space of five points in the real projective line II" (in preparation).

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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