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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

令Y和Y'为n维球体中的一对曲线段，并令x、x+dx为Y上的点，且y、y+dy为Y'上的点。当我们总是假设 x 和 y 不同时，我们允许 Y 和 Y' 重合的情况。通过球极投影将x、x+dx、y、y+dy这四个点与黎曼球识别为一个二维球体，得到这四个点的交比。那么它可以被认为是YxY'上的复值2-形式。我们称之为 Y 和 Y' 的无穷小交叉比。根据定义，它在莫比乌斯变换下是不变的。我们获得了对无穷小交比的实部和虚部的新解释。n维球体可以被实现为(n+2)维明可夫斯基空间中无穷大光锥的点集。设S(n,p)表示n维球体中p维球体的集合。那么 S(n, p) 可以用 Pluecker 坐标表示，是一个具有不定 m … 更多 etric 的空间。 n维球体中的一对曲线段Y和Y'也可以被认为是S(n,O)中的一个曲面。现在无穷小交叉比的实部等于该表面的面积元素的绝对值。另一方面，虚部的解释可以如下给出。 n维球体可以被认为是(n+1)维双曲空间的边界。令 L 表示双曲空间中连接边界球中 Y 上的点 x 和 Y' 上的 y 的测地线，并令 P 为 L 的正交超平面。令 x' 和 y' 分别为 x 和 y 邻域中的点。 P 与连接 x' 和 y' 的测地线的交点给出了 P 中的一个曲面。然后 (x, y) 处的无穷小交叉比的虚部等于该曲面的面积元素。我们还使用空间 S(n, p) 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 Jun Imai 和 Remi 共同工作的总结 2004 年，今井在法国逗留 7 个月期间，担任海外调查员的朗之万 (Langevin)。2003 年，部分补助金用于邀请朗之万 (Langevin) 来日本。）