1994 Fiscal Year Final Research Report Summary
GEOMETRIC STUDY OF VARIATIONS ON INFINITE DIMENSIONAL MANIFOLDS
| Project/Area Number |
05452006
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Geometry
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| Research Institution | THE UNIVERSITY OF TOKYO |
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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)
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| 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)
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| Project Period (FY) |
1993 – 1994
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| Keywords | GEOMETRIC VARIATIONAL PROBLEM / DYNAMICAL SYSTEMS / MORSE THEORY |
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| 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
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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
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