| Project/Area Number |
20540218
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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 | Tokyo University of Science |
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Principal Investigator |
FURUTANI Kenro Tokyo University of Science, 理工学部, 教授 (70112901)
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| Co-Investigator(Renkei-kenkyūsha) |
IWASAKI Chisato 兵庫県立大学, 大学院・物質理学研究科, 教授 (30028261)
MORIMOTO Toru 奈良女子大学, 理学部, 教授 (80025460)
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| Project Period (FY) |
2008 – 2010
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| Project Status |
Completed (Fiscal Year 2010)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2008: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | sub-Riemann多様体 / sub-Laplacian / 熱核 / spectral zeta function / zeta-regularized determinant / ベキ零リー群 / Grushin type 作用素 / non-holonomic sub-bundle / Toeplitz作用素 / 幾何学的量子化 / non-holonomic subbundle / sub-Riemannian多様体 / heat Kernel / Grushin type operator / Grushin type Operator / Hamilton system / 陪特性曲線 / nilmanifold / Berezin変換 / 再生核 / sub-Laplaciar / Hankel 作用素 / Toepl〓作用素 |
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| Research Abstract |
Mainly we focused our study on the topics on analytic and geometric aspects of sub-Riemannian manifolds. Our sub-Riemannian manifolds are in the strong sense, that is、the sub-Riemannian structure is defined by a non-holonomic sub-bundle which is trivial as a vector bundle. So we have a sub-elliptic operator on such manifolds, which we call sub-Laplacian. Examples of such manifolds are three and seven dimensional spheres, or nilmanifolds and we studied their spectral zeta functions from the point of views of analytic continuation, residue calculus and the explicit determination of zeta regularized determinant. Also, we investigate a general framework to define a sub-elliptic operator from a sub-Laplacian on the total space to the base manifold through a submersion (or more specifically fiber bundle setting) and obtained a relation between their bi-characteristic flows. Especially we find a relation between sub-Riemannian geodesics on S^3 through Hopf fiber bundle S^3→CP^1 by notifying their relation with the isoperimetric aspect of curves on CP^1 and by considering double fiberings defined by left and right quaternion multiplication structure on S^3, we determine bi-characteristic flow of the spherical Grushin operator.
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