2006 Fiscal Year Final Research Report Summary
Energy of knots and conformal geometry
| Project/Area Number |
17540089
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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 | Tokyo Metropolitan University |
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Principal Investigator |
IMAI Jun Tokyo Metropolitan University, Graduate School of Science and Technology, Associate professor, 大学院理工学研究科, 准教授 (70221132)
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| Co-Investigator(Kenkyū-buntansha) |
AKAHO Manabu Tokyo Metropolitan University Graduate School of Science and Technology, Graduate School of Science and Technology, Research Associate, 大学院理工学研究科, 助手 (30332935)
YOKOTA Yoshiyuki Tokyo Metropolitan University Graduate School of Science and Technology, Graduate School of Science and Technology, Associate professor, 大学院理工学研究科, 助教授 (40240197)
KAMISHIMA Yoshinobu Tokyo Metropolitan University Graduate School of Science and Technology, Graduate School of Science and Technology, Professor, 大学院理工学研究科, 教授 (10125304)
GUEST Martin Tokyo Metropolitan University Graduate School of Science and Technology, Graduate School of Science and Technology, Professor, 大学院理工学研究科, 教授 (10295470)
OHNITA Yoshihiro Osaka City University, Dept. of Math, Professor, 理学研究科, 教授 (90183764)
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| Project Period (FY) |
2005 – 2006
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| Keywords | topology / knot theory / energy / conformal geometry |
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| Research Abstract |
Conformal Geometry of Curves and Surfaces (Joint work with Remi Langevin, who is an investigator abroad):
Let x and y be points on a curve C in the 3 dimensional sphere. We can define a complex valued 2-form on ac-t by first identifying the sphere through four points x, x+dx, y, and y+dy with the Riemann sphere CU {∞} and then by taking the cross ratio of the four complex numbers corresponding to the Jour points through a stereographic projection. Let us call it the infinitesimal cross ratio. It is, by definition, invariant under Moebius transformations.
The real and the imaginary parts of it can be interpreted as follows.
Let S(p,3) denote the set of oriented p dimensional spheres in the 3-sphere. We can give a pseudo-Riemannian structure on it by using Pluecker coordinates. The space CxCΔ can be considered a surface in S(0,3). The real part of the infinitesimal cross ratio is equal to a signed area element of this surface.
The space S(0,3) also admits a symplectic structure as the cotangent bundle of 3-sphere. The real part of the infinitesimal cross ratio is also equal to the canonical symplectic form of S(0,3).
Topology of planar linkages :
The configuration space of the planar mechanism of a robot with $n$ anus each of which has a rotational joint and a fixed end point is studied. Its topological type is given by a Morse theoretical way and a topological way.
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Research Products
(5 results)
