Energy of knots and conformal geometry

设Y和Y'是n维球面上的一对曲线段，x，x+dx是Y上的点，y，y+dy是Y'上的点。当我们总是假定x和y是不同的时，我们允许Y和Y'重合的情况。通过球极投影将通过x、x+dx、y和y+dy四个点的二维球体与黎曼球体识别出来，我们就可以获得这四个点的交比。则它可以被认为是YxY '上的复值2-形式。我们称之为Y和Y '的无穷小交比。我们得到了无穷小交比的真实的部分和虚部的新解释.一个n维球面可以表示为（n+2）维Minkowski空间中光锥的无穷远点集.设S（n，p）表示n维球面中p维球面的集合.则S（n，p）是具有不定m的空间，可用Pluecker坐标表示 关于我们 电的在n维球面上的一对曲线段Y和Y'也可以被认为是S（n，O）中的曲面。现在无穷小交比的真实的部分等于这个曲面的面积元素的绝对值。另一方面，虚部的解释可以给出如下。一个n维球面可以被认为是（n+1）维双曲空间的边界。设L表示双曲空间中连接边界球面中Y上的点x和Y'上的点y的测地线，P是与L正交的超平面。设x'和y'分别是x和y的邻域中的点。P与连接x'和y'的测地线的交给出P中的一个曲面，则在（x，y）处的无穷小交比的虚部等于该曲面的面积元.我们还利用空间S（n，p）上的共形不变测度定义了曲线曲面空间上的泛函. (That是Jun Imai（2003年担任首席调查员）和Remi Langevin（2004年担任海外调查员）在Imai于法国停留7个月期间的联合工作总结。2003年，部分赠款用于邀请伦杰万访问日本。少

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