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2002 Fiscal Year Final Research Report Summary

Global Studies on Curvature and Geometric Structures of Riemannian Manifolds

Research Project

Project/Area Number 13640093
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionFukuoka Institute of Technology

Principal Investigator

ITOKAWA Yoe  Fukuoka Institute of Technology, Faculty of Information Engineering, Professor, 情報工学部, 教授 (90223205)

Co-Investigator(Kenkyū-buntansha) NISHIHARA Masaru  Fukuoka Institute of Technology, Faculty of Information Engineering, Professor, 情報工学部, 教授 (20112287)
SHIRAKAWA Hiroshi  Fukuoka Institute of Technology, Faculty of Information Engineering, Professor, 情報工学部, 教授 (60002995)
GOTO Midori  Fukuoka Institute of Technology, Faculty of Information Engineering, Professor, 情報工学部, 教授 (60162161)
NISHIYAMA Takahiro  Fukuoka Institute of Technology, Faculty of Information Engineering, Assistant Professor, 工学部, 助教授 (60333241)
Project Period (FY) 2001 – 2002
Keywordsriemannian manifolds / sectional curvature / comparison theorems / locally pseudoconvex spaces / extension of functions / Hahn-Banach type theorems / Euler eq / Galerkin approximation
Research Abstract

Head investigator, Itokawa in the papers "Maximal diameter theorem for manifolds with restricted radial curvature" and "Generalized Toponogov's theorem for manifolds with radial curvature bounded below" written jointly with Katsuhiro Shiohama of Saga University and Yoshiro Machigashira of Osaka University of Educations, investigated riemannian manifolds whose sectional curvature in radial directions from a fixed point is bounded from below by certain function. We have obtained some triangle comparison theorems generalizing that of Toponogov. Moreover, in case equality is attained in these comparisons, these manifolds exhibit very rigid geometrical structure containing some totally geodesic surfaces. We have also obtained some applications of these results including a new sphere theorem.
Investigator Nishihara is continuing his study on the extendibility of certain functions on infinite dimensional locally pseudo-convex spaces. In "The extension of polynomials of integral type in locally … More convex spaces and its applications" and "The extension of entire functions of nuclear type on locally convex spaces", he has extended his own results on the extendibility to the total space of polynomials and holomorphic functions which are a priori only defined on locally convex spaces. In the paper "The extension of entire function of nuclear type", he has also succeeded in generalizing Hahn-Banach type theorems of Meise-Vogt and of Nishihara himself. Further extension will be discussed in a forthcoming paper "A Hahn-Banach extension theorem for entire functions of nuclear type". In another forthcoming paper "Pseudoconvex domains of infinite dimensional Grassmann manifolds", Nishihara has also proved a vanishing theorem for regular pseudoconvex domains in Banach spaces.
Nishiyama who has substituted as an investigator for the year 2001 has written two papers "Pseudo-advection methods for the axisymmetric stationary Euler equations" and "Magnetohydrodynamic approach to the solvability of the three-dimensional stationary Euler equations". In these papers, Nishiyama studied the methods for obtaining stationary solutions to certain Euler type equations. In particular, he established the effectiveness of the method of using the Galerkin approximation on certain accompanying equations associated to the given equation in case the given equation has a rotationary symmetry around the axis and for general equations in dimension 3. Less

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Yoe Itokawa: "Maximal diameter theorems for manifolds with restricted radial curvature"Proceedings of the Fifth Pacific Rim Geometry Conference. 61-68 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yoe Itokawa: "Generalized Toponogov theorem for manifolds with radial curvature bounded below"Robert E. Greene 教授60歳記念論集. (発刊予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masaru Nishihara: "The extension of polynomials of integral type in locally convex spaces and its applications"Proceedings of the Eighth International Colloquium on Finite or Infinite Dimensional Complex Analysis. 161-165 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masaru Nishihara: "The extension of entire functions of nuclear type on locally convex spaces"Proceedings of the Ninth international Colloquium on Finite or Infinite Dimensional Complex Analysis. (発刊予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takahiro Nishiyama: "Pseudo-advection method for the axisymmetric stationary Euler equations"Zeitschrifte fur Angewandte Ilathematik und Mechanik. 81. 711-715 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takahiro Nishiyama: "Magnetohydrodynamic approach to solvability of the three-dimensional stationary Euler equations"Glasgow Uathematical Journal. (発刊予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yoe Itokawa, Yoshiro Machibashira, Kathuhiro Shiohama: "Maximal diameter theorems for manifolds with restricted radial curvature"Proceedings of the Fifth Pacific Rim Geometry Conference. 61-68 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoe Itokawa, Yoshiro Machigashira, Katsuhiro Shiohama: "Generalized Toponogov theorem for manifolds with radial curvature bounded below"Collected Papers in Honor of Professor Robert E. Green's 60th. Birthday. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masaru Nishihara: "The extension of polynomials of integral type in locally convex spaces and its applications"Proceedings of the Eighth International Colloquium on Finite or Infinite Dimensional Complex Analysis. 161-165 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masaru Nishihara: "The extension of entire functions of nuclear type on locally convex spaces"Proceedings of the Ninth International Colloquium on Finite or Infinite Dimensional Complex Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takahiro Nishiyama: "Pseudo-Advection methods for the stationary Euler equations"Zeitschrifte fur Angewandte Mathematik und Mechanik. 81. 711-715 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takahiro Nishiyama: "Magnetohydrodynamic approach to solvability of the three-dimensional stationary Euler equations"Glasgow Mathematical Journal. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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