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GEOMETRIC STUDY OF VARIATIONS ON INFINITE DIMENSIONAL MANIFOLDS

Research Project

Project/Area Number 05452006
Research Category

Grant-in-Aid for General Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionTHE UNIVERSITY OF TOKYO

Principal Investigator

OCHIAI Takushiro  GRADUATE SCHOOL OF MATHEMATICAL SCIENCES,THE UNIVERSITY OF TOKYO,PROFESSOR, 大学院・数理科学研究科, 教授 (90028241)

Co-Investigator(Kenkyū-buntansha) ISHIMURA Naoyuki  GRAD.SCH.OF MATH.SCI., THE UNIVERSITY OF TOKYO,ASSISTANT, 大学院・数理科学研究科, 助手 (80212934)
OTSU Yukio  GRAD.SCH.OF MATH.SCI., THE UNIVERSITY OF TOKYO,ASSISTANT, 大学院・数理科学研究科, 助手 (80233170)
TSUBOI Takashi  GRAD.SCH.OF MATH.SCI., THE UNIVERSITY OF TOKYO,ASSO.PRO., 大学院・数理科学研究科, 助教授 (40114566)
MATANO Hiroshi  GRAD.SCH.OF MATH.SCI., THE UNIVERSITY OF TOKYO,PRO, 大学院・数理科学研究科, 教授 (40126165)
MATSUMOTO Yukio  GRAD.SCH.OF MATHEMATICAL SCI., THE UNIVERSITY OF TOKYO,PRO., 大学院・数理科学研究科, 教授 (20011637)
深谷 賢治  東京大学, 大学院・数理科学研究科, 助教授 (30165261)
砂田 利一  東北大学, 理学部, 教授 (20022741)
Project Period (FY) 1993 – 1994
Project Status Completed (Fiscal Year 1994)
Budget Amount *help
¥6,400,000 (Direct Cost: ¥6,400,000)
Fiscal Year 1994: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1993: ¥3,600,000 (Direct Cost: ¥3,600,000)
KeywordsGEOMETRIC VARIATIONAL PROBLEM / DYNAMICAL SYSTEMS / MORSE THEORY / 幾何学的変分問題 / 危点 / 調和写像 / 極小曲面 / シンプレクティク幾何学 / 物理幾何学 / モース関数 / 量子化 / 大域変分法 / 測地線 / 偏微分方程式
Research Abstract

We have studied some geometry on infinite dimensional manifolds. Especially we have studied manifolds given by some functional spaces on finite dimensional manifolds, and as results we have obtained important contribution to study topological properties of the original manifolds.
More precisely, we have studied various functionals derived from finite dimensional manifolds and their variations. The main idea behind this setting is to generalize the Morse theory well studied so far. We have reduced the study of Morse theory to that of dynamical systems through the gradient vector fields. Thorough this view point, we have been able to obtain some contribution to the study of infinite dimensional dynamical systems. As results of our methods, we have been able to clarify some of topological and geometrical properties of infinite and finite dimensional manifolds desclibed as follows.
At first we studied some topology of the space of Riemannian metrics on finite dimensional manifolds. We have d … More one this study from three view points explained as follows. The first view point is to study the variations of the several (integral geometric) invariants derived from metrics. The critical values of these invariants are given as solutions of various partial differetial equations. So we have been able to adopt the method of various problems and partial differetial equations. The second view point is to consider the limits of various geometric inequaliteis derived from the curvatures of Riemannian metrics. Although many of the limits of these inequalities cannot be characteraized from partial differential equations, we have found that instead usual Riemannian geometric methods are usufull. The third view point is to observe that the spaces concered are the classifying spaces of (infinite dimentional) Lie groups of the diffeomorphisms of manifolds. Therefore we have found that the usual methods in the study of the classifying spaces of foliated product are very useful.
Moreover we have studied the topology of the space of the moduli of the given geometric structures on finite dimensional manifolds, and that of the family of smooth mappings between two finite dimentional manifolds. This problem contains the study of submanifolds. For these problems, we have taken the method of variations of various functionals. Less

Report

(3 results)
  • 1994 Annual Research Report   Final Research Report Summary
  • 1993 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Matsumoto.Y: "On the topological structure of the Fermat surface of degree 5" Kodai Math.J. 17. 560-570 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Matano.H: "Nonlinear partial defferential eg : and infinile dim.dynamical cystem" Amer.Math.Soc.Tvans (2). 161. 1-17 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Tsuboi.T: "Homological and dynamical study on certain groups" J.Math.Soc.Japan. 47. 1-30 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Otsu.S: "The Riewannian structure of Alexandrov spares" J.Diff.Geom. 39. 629-658 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Ishimura.N: "Inertial manifolds for Burgirs' onginal sys. of turbuleuce" Appl.Math.Lett. 7-3. 33-37 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Ochiai.T: "Introduction to Differential Geometry II" Univ.of Tokyo Press, 225 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Matsumoto, Y: "On the topological structure of the Fermat surface of degree 5" Kodai Math.J.17. 560-570 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Matano, H: "Nonlinear partial differential equations and infinite dimensional dynamical systems" Amer.Math.Soc.Trans.(2) 161. 1-17 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Tsuboi, T: "Homological and dynamical study on certain groups" J.Math.Soc.Japan. 47. 1-30 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Otsu, S: "The Riemannian structure of Alexsandrov spaces" J.Diff.Geom. 39. 629-658 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Ishimura, N: "Inertial manifolds for Burgero original systems of turbulance" Appl.Math.Lett.7-3. 33-37 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Ochiai, T: introduction to Differential Geometry II. Univ.of Tokyo Press, 225 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Yukio Matsumoto: "On the topological structure of the Fermat surface of degree5" Kodai Math.J.17. 560-570 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] C.M.Elliott,H.Matano and T.Oi: "Zeros of a complex Ginzburg-Landau order parameter with applications to superconductivity" European Journal of Applied Mathematics. 5. 431-448 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Tsuboi: "Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle" J.Math.Soc.Japan. (発表予定).

    • Related Report
      1994 Annual Research Report
  • [Publications] N.Ishimura,I.Ohnishi: "Inertial manifolds for Burgers´ original model system of turbulence" Appl.Math.Letters. 7-3. 33-37 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] Y.Otsu,T.Shioya: "The Riemannian structure of Alexandrov spaces" J.Differential Geom.39. 629-658 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] Y.Matsumoto,J.M Montosinus: "Pseudo Beriodic Humeumurphisms and degeneration of Piipnar Swifaces" Bull AMS.(発表予定). (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] Toshiaki Adachi,Toshikazu Sunada: "Density of Hates in Syectial glimetry" Commentary Mathematics Helvetic.68. 480-493 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] Fukaya Kenji,Yamaguchi Takao: "Isometry group of Singular spaces" Mathematisches Zeitschrift. (発表予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] Fukaya Kenji: "Floer homology for 3 manifolds with boundary" Proceeding of Taniguchi Symposium(World Scientific). (予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] Fukaya Kenji: "Floer homology for connected sum of homology 3-spheres" Topology. (発表予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] Ootsu Yukio: "The Riemannicm Structure of Alexandrov Spaces" Journal of Differential Geometry. (発表予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] 落合卓四郎: "微分幾何学入門" 東大出版会, 500 (1993)

    • Related Report
      1993 Annual Research Report

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Published: 1993-04-01   Modified: 2016-04-21  

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