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Energy of knots and conformal geometry

结的能量和共形几何

基本信息

批准号：
17540089
负责人：
IMAI Jun
金额：
$ 1.79万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540089/
关键词：
topology knot theory energy conformal geometry

项目摘要

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.

曲线和曲面的共形几何(与国外研究员雷米·朗之万共同工作)：设x和y是三维球面上曲线C上的点。首先用黎曼球面CU{∞}通过四个点x，x+dx，y，y+dy识别球面，然后通过赤平投影求对应于这四个点的四个复数的交比，我们就可以在Ac-t上定义一个复值2-形式。让我们称它为无穷小交叉比。根据定义，它在Moebius变换下是不变的。它的实部和虚部可以解释如下。设S(p，3)表示三维球面中定向的p维球面的集合。利用Pluecker坐标可以给出其上的伪黎曼结构。空间CxCΔ可以看作是S(0，3)中的一个曲面。无穷小交叉比的实部等于该曲面的符号面积元，S(0，3)空间也允许辛结构为3-球面的余切丛。无穷小交叉比的实部也等于S(0，3)的典范辛形。平面连杆机构的拓扑：研究了一个具有转动关节和固定端点的具有$n$肛门的机器人平面机构的位形空间。用Morse理论和拓扑学方法给出了它的拓扑类型。