| Project/Area Number |
11640073
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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 |
Geometry
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| Research Institution | Nagoya Institute of Technology |
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Principal Investigator |
ADACHI Toshiaki Nagoya Institute of Technology, Faculty of Engineering, Professor, 工学部, 助教授 (60191855)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAGISHI Masakazu Nagoya Institute of Technology, Faculty of Engineering, Associated Professor, 工学部, 助教授 (40270996)
OHYAMA Yoshiyuki Nagoya Institute of Technology, Faculty of Engineering, Associated Professor, 工学部, 助教授 (80223981)
MAEDA Sadahiro Shimane University, Interdisciplinary, 総合理工学部, 教授 (40181581)
EJIRI Norio Nagoya Institute of Technology, Faculty of Engineering, Professor, 工学部, 教授 (80145656)
OHTSUKA Fumiko Ibaraki University, Faculty of Science, Associated Professor, 理学部, 助教授 (90194208)
山本 和広 名古屋工業大学, 工学部, 教授 (30091515)
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| Project Period (FY) |
1999 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2000: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Kaehler magnetic field / length spectrum / geodesic sphere / circle / complex space form / shperical mean / helix / Kaehler immersion / Kaehler immersion / Veronese emmbeding / curve of order 2 / kahler magnetic field / closed geodesic / geodesis sphere |
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| Research Abstract |
The head investigator studied trajectories for Kaehler magnetic fields on symmetric spaces. His works, a part of which is a joint work with some coinvestigators, can be classified into the following four directions.
(1) Mean operators associated with magnetic fields
In order to study the relationship between trajectories for Kaehler magnetic fields and Schroedinger operators, the head investigator studied magnetic random walks. On complex space forms the mean operator generated by this magnetic random walk has the same properties as that generated by the geodesic random walk. On the other hand, if we studied a magnetic spherical mean which is derived from potential on unit tangent bundle we found the principal term of the formal expansion was the Schroedinger operator.
(2) Circles on complex space forms
Extending the notion of trajectories for Kaehler magnetic fields the head investigator studied length spectrum for circles on complex space forms. The moduli space of circles on these space
… More
s are open rectangles in a Euclidean plane parametrized by geodesic curvature and complex torsion. Concerning the continuity of length spectrum for circles we found it had a natural foliation structure. By use of this structure we clearfied set theoretic properties of length spectrum, the asaymptotic behavior of the number of congruency classes of circles with respect to their length, and the properties of the k-th length spectrum function with respect to the geodesic curvature.
(3) Geodesics on geodesic spheres in a rank one symmetric spaces
Having been inspired with the idea in our study on circles we studied lengths of geodesics on a geodesic sphere in a rank one symmetic space, which is famous as an example of Berger sphere. We considered geodesics on a geodesic sphere as curves on a complex space form, and studied their horizontal lifts with respect to the Hopf fibration. We could then treat them as curves in a Euclidean space. We showed the relationship between the radius of a geodesic sphere and length-simplicity of geodesics and clearfied the asymptotic behavior of the number of closed geodesics on a geodesic sphere with respect to their length.
(4) Characterizations of submanifolds in complex space forms
By use of properties of circles and helices on complex space forms the head investigator and S. Maeda characterized submanifolds in complex space forms. Their idea stands on the technique of treating geodesic and circles on a submanifold as curves on a complex space form. They characterized homogeneous submanifold, Veronese embeddings and some other important submanifolds. Less
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