2000 Fiscal Year Final Research Report Summary
Studies of Analysis on manifolds
| Project/Area Number |
10640214
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Science University of Tokyo |
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Principal Investigator |
FURUTANI Kenro Science university of Tokyo, Faculty of Science of Technology Professor, 理工学部, 教授 (70112901)
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| Co-Investigator(Kenkyū-buntansha) |
KOBAYASHI Reido Science university of Tokyo, Faculty of Science and Technology Professor, 理工学部, 教授 (70120186)
OKA Masatoshi Science university of Tokyo, Faculty of Science and Technology Professor, 理工学部, 教授 (70120178)
OTSUKI Nobukazu Science university of Tokyo, Faculty of Science and Technology Professor, 理工学部, 教授 (80112895)
TANAKA Makiko Science university of Tokyo, Faculty of Science and Technology Lecturer, 理工学部, 講師 (20255623)
KOBAYASHI Takao Science university of Tokyo, Faculty of Science and Technology associated Professor, 理工学部, 助教授 (90178319)
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| Project Period (FY) |
1998 – 2000
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| Keywords | Cayley projective plane / Kahler form / Maslov index / Spectral flow / geodesic flow / quantization / Fredholm pair / elliptic operator |
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| Research Abstract |
1. We developed a theory of Maslov index in the infinite dimension. In the finite dimension we have a principal bundle U(n)→Λ(n) = U(n)/O(n), also we have a principal bundle similar to that in the infinite dimension, but it is trivial by Kuiper theorem. Fortunately we have a slightly different principal bundle with the total space not being a group and we call its base space, Fredholm Lagrangian Grassmannian. Especially we found a functional analytic method to define a Maslov index for not only loops but also for arbitrary paths with respect to a fixed Maslov cycle and we applied this theory to generalize Yoshida-Nicolaescu theorem for the spectral flow of a family of first order self-adjoint elliptic operators on a closed manifold on the framework of the theory of self-adjoint extensions of symmetric operators. Relating with the study of Fredholm Lagrangian Grassmannian we prove that the homotopy groups of the space of unitary operators of the form "U + Id is Fredholm" is isomorphic to the stable homotopy groups of unitary groups.
2. We constructed a Kahler structure on the punctured cotangent bundle of the Cayley projective plane and embedded it into a space of 8×8 complex matrices. Then we give an explicit representation of the geodesic flow of the Cayley projective plane in terms of this embedding.
3. By making use of a Kahler structure on the punctured cotangent bundle of the quaternion projective space P^nH, we construct a Hilbert space consisting of certain classes of holomorphic functions with the reproducing kernel, and also constructed an operator from this Hilbert space to L_2(P^nH), and proved that it represents the geodesic flow in terms of the one parameter family of elliptic Fourier integral operators generated by √<Δ+(2n+1)_2> (quantization of the geodesic flow).
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