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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：
15540088
负责人：
OHNITA Yoshihiro
金额：
$ 2.24万
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：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

Let Y and Y' be a pair of curve segments in an n dimensional sphere, and let x, x+dx be points on Y and y, y+dy be points on Y'. We allow the case when Y and Y' coincide, when we always assume that x and y are distinct. By identifying a 2 dimensional sphere through the four points x, x+dx, y, and y+dy with the Riemann sphere through a stereographic projection, we obtain the cross ratio of these four points. Then it can be considered as a complex valued 2-form on YxY'. We call it the infinitesimal cross ratio of Y and Y'. It is, by definition, invariant under Moebius transformations.We obtained new interpretations of the real and the imaginary parts of the infinitesimal cross ratio.An n dimensional sphere can be realized as the set of points a infinity of the light cone in (n+2) dimensional Minkowski space. Let S(n, p) denote the set of p dimensional sphere in the n dimensional sphere. Then S(n, p), which can be expressed in terms of Pluecker coordinates, is a space with an indefinite m … More etric. A pair of curve segments Y and Y' in the n dimensional sphere can also be considered as a surface in S(n, O). Now the real part of the infinitesimal cross ratio is equal to the absolute value of the area element of this surface.On the other hand, the interpretation of the imaginary part can be given as follows. An n dimensional sphere can be considered as the boundary of the (n+1) dimensional hyperbolic space. Let L denote a geodesic in the hyperbolic space joining points x on Y and y on Y' in the boundary sphere, and let P be an orthogonal hyperplane to L. Let x' and y' be points in neighborhoods of x and y respectively. The intersection of P and the geodesic joining x' and y' gives a surface in P. Then the imaginary part of the infinitesimal cross ratio at (x, y) is equal to the area element of this surface.We also defined functionals on the space of curves and surfaces using a conformally invariant measure on the space S(n, p).(That is the summary of the joint work of Jun Imai, who was the head investigator in 2003, and Remi Langevin, who is an investigator abroad, during Imai's 7 months stay in France in 2004. Part of the grant was used to invite Lengevin to Japan in 2003.) Less

设Y和Y'是n维球面上的一对曲线段，设x x+dx是Y上的点Y +dy是Y上的点。我们允许Y和Y重合的情况，当我们总是假设x和Y是不同的。通过对黎曼球进行立体投影，通过x、x+dx、y、y+dy四个点识别一个二维球面，得到这四个点的交比。则可以认为它是YxY'上的复值2型。我们称它为Y和Y'的极小交叉比。根据定义，它在莫比乌斯变换下是不变的。我们得到了无限小交叉比的实部和虚部的新解释。一个n维球体可以被实现为光锥在（n+2）维闵可夫斯基空间中的无穷点的集合。设S（n, p）表示n维球面中p维球面的集合。S（n, p）可以用Pluecker坐标表示，它是一个不确定的空间。n维球面上的一对曲线段Y和Y'也可以看作S（n， O）中的一个曲面。现在，这个极小交叉比的实部等于这个曲面的面积元的绝对值。另一方面，虚部的解释可以如下所示。一个n维球面可以看作是（n+1）维双曲空间的边界。设L表示双曲空间中的测地线，将边界球上的点x与Y上的点Y连接起来，设P是与L正交的超平面，设x′和Y′分别是x和Y的邻域内的点。P与x‘和y’的测地线相交，在P中得到一个曲面，那么在（x, y）处的无限小交叉比的虚部就等于这个曲面的面积元。我们还利用S（n, p）空间上的共形不变测度定义了曲线和曲面空间上的泛函。（这是2003年首席研究员今井俊和国外研究员Remi Langevin于2004年在法国7个月期间共同工作的总结。）这笔资金的一部分用于邀请兰格文2003年访问日本。)少