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Geometry of Points at Infinity

Research Project

Project/Area Number 12640069
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka Kyoiku University

Principal Investigator

SUGAHARA Kunio  Osaka Kyoiku U., Faculty of Education, Prof., 教育学部, 教授 (20093255)

Co-Investigator(Kenkyū-buntansha) MACHIGASHIRA Yoshiroh  Osaka Kyoiku U., Faculty of Education, Asso. Prof., 教育学部, 助教授 (00253584)
KOYAMA Akira  Osaka Kyoiku U., Faculty of Education, Prof., 教育学部, 教授 (40116158)
KATAYAMA Yoshikazu  Osaka Kyoiku U., Faculty of Education, Prof., 教育学部, 教授 (10093395)
ITOH Jin-ichi  Kumamoto U., Faculty of Education, Prof., 教育学部, 教授 (20193493)
INNAMI Nobuhiro  Niigata U., Faculty of Science, Prof., 理学部, 教授 (20160145)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsRiemannian manifold / Liouville manifold / ideal boundary / Alexandrov空間 / Busemann関数
Research Abstract

For a Riemannian manifold there are several definitions of points at infinity. Gromov defined points at infinity using only the metric structure of the Riemannian manifold and named it ideal boundary.
Although his definition is abstract and universal, the relation between the global Riemannian structure and the ideal boundary is not clear. And it is hard to determine the ideal boundary for each Riemannian manifold. In fact, in the global study of Hadamard manifolds, he did not essentially make use of the idea of the ideal boundary.
In this research, in order to study the geometry of the ideal boundary, we tried to determine the ideal boundary for some Riemannian manifolds.
The quadratic surfaces in Euclidean space are Liouville manifolds. Making use of their classical coordinates, we studied the differential equation of their geodesies and see that the points at infinity of elliptic paraboloids are determined by the limit of the distance between two singular sets as Liouville manifolds.
For the quadratic surfaces in hyperbolic space, we can use the analogous method in low dimensions. Hyperboloids of two sheets has finite Maeda constant and the same geodesic structure with elliptic paraboloids in Euclidean spaces and their points at infinity are also determined by the limit of the distance between two singular sets as Liouville manifolds.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (25 results)

All Other

All Publications (25 results)

  • [Publications] Hangan, T.: "Acute triangulations"Bull.Math.Soc.Sci.Math.Roumanie. Vol.43. 279-286 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Itoh, J.: "The Lipschitz continuity of the distance function to the cut locus"Transactions of American Mathematical Society. Vol.353. 21-40 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Katayama, Y.: "Characteristic invariant of tensor productactions and actions on crossed product"J.Austral.Math.Soc.Ser.A. Vol.18(to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Machigashira, Y.: "Total excess on length surfaces"Math. Ann.. Vol.319. 675-706 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Itoh, J.: "Essential cut locus of on a surface"Tohoku Math.Publ.. Vol.20. 53-59 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Nobuhiro, I.: "Gradient vector fields which characterize warped products"Math. Scand.. Vol.88. 182-192 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Nobuhiro, I.: "Geometry of geodesics for convex billiards and circular billiards"Math.J.. Vol.13. 73-120 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Koyama, K.: "Cohomological dimension and acyclic resolutions"Topology and its Appl.. Vol.120. 175-204 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Nobuhiro, I.: "Geometry of geodesics for convex billiards and circular billiards"Nihonkai Math.J.. Vol.13. 73-120 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hangan, T.: "Acute triangulations"Bull. Math. Soc. Sci. Math. Rournanie. Vol.43. 279-286 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Itoh, J.: "The Lipschitz continuity of the distance function to the cut locus"Transactions of American Mathematical Society. Vol.353. 21-40 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Katayama. Y.: "Characteristic invariant-of tensor productactions and actions on crossed product"to appear in J. Austral. Math. Soc. Ser. A. Vol.18.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Machigashira, Y.: "Total excess on length surfaces"Math. Ann.. Vol.319. 675-706 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Itoh, I.: "Essential cut locus of on a surface"Tohoku Math. Publ.. Vol.20. 53-59 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Nobuhiro, I.: "Gradient vector fields which characterize warped products"Math. Scand.. Vol.88. 182-192 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Koyama, K.: "Cohomological dimension and acyclic resolutions"Topology and its Appl.. Vol.120. 175-204 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Noibuhiro. I.: "Geometry of geodesies for convex billiards and circular billiards"Nihonkai Math. J.. Vol.13. 73-120 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akira Koyama, Katsuya Yokoi: "Cohomological dimension and acyclic resolutions"Topology and its Appl.. Vol.120. 175-204 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nobuhiro, I.: "Geometry of geodesies for convex billiards and circular billiards"Nihonkai Math. J.. Vol.13. 73-120 (2002)

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

    • Related Report
      2001 Annual Research Report
  • [Publications] Itoh, J.: "Essential cut locus of on a surface"Tohoku Math.Publ.. Vol.20. 53-59 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Nobuhiro, I.: "Gradient vector fields which characterize warped products"Math. Scand.. Vol.88. 182-192 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hangan,T.: "Acute triangulations"Bull.Math.Soc.Sci.Math.Roumanie. Vol.43. 279-286 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Itoh,J.: "The Lipschitz continuity of the distance function to the cut locus"Transactions of American Mathematical Society. Vol.353. 21-40 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Katayama,Y.: "Characteristic invariant of tensor productactions and actions on crossed product"to appear in J.Austral.Math.Soc.Ser.A. Vol.18.

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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