Convergence of curvatures in secant approximations
Convergence of curvatures in secant approximations
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割线近似中曲率的收敛
DOI：
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
Joseph H. G. Fu
中科院分区：
文献类型：
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作者：
Joseph H. G. Fu
很久以前就知道，欧氏空间E中的闭多面体P允许某些类似于经典曲率积分的曲率测度（参见图1）。[3]，[11]，[17]，[1]，[15]）。如果P1，P，. . .是收敛到Mc E的光滑子流形的一列这样的多面体，自然要问P的曲率测度是否收敛到M的相应曲率积分。(In著名的面积理论的例子（参见）。[16，1.1.10]）当然，在精确地表述假设时，必须小心。）这个方程的内在类似物已经由Cheeger、M. M. M. M. Ller和Schrader在[5]中给出了肯定的回答，他们还断言，他们的方法同样适用于上面的外在问题。本文的目的是给出一个比文献[5]的解在概念上简单得多的外问题的解。我们的方法基于这样一个观察：E中多面体P（或光滑子流形M）的曲率测度（或积分）可以从与P（或M）正则相关的某个积分流以通用的方式计算，该积分流存在于切球丛SE = E × S~（cf. [19]，[20]，[6]）。如果M是光滑的，则该电流通过在M的单位法线的正则定向的（n l）-流形N（M）上积分给出。我们可以把一个类似的物体N（P）与一个多面体P联系起来，尽管N（P）不再是SE的子流形，而是一个维数为n1的积分流，称为P的法圈。为了得到P的曲率测度，我们观察到在SE n中存在普适的微分（n1）形式κ0，，κn_{，使得P的曲率测度由下式给出：
It has long been known that a closed polyhedron P in Euclidean space E admits certain curvature measures analogous to classical curvature integrals (cf. [3], [11], [17], [1], [15]). If P 1 , P, . . . is a sequence of such polyhedra converging to a smooth submanifold of M c E , it is natural to ask whether the curvature measures of the P converge to the corresponding curvature integrals of M. (In view of well-known examples in area theory (cf. [16,1.1.10]) it is of course necessary to take some care in formulating the hypothesis precisely.) An intrinsic analogue of this equation has been answered positively in [5] by Cheeger, Mϋller and Schrader, who have also asserted that their method applies equally well to the extrinsic question above. Our aim in the present article is to give a solution to the extrinsic problem that is conceptually much simpler than the solution of [5]. Our approach rests on the observation that the curvature measures (or integrals) of polyhedra P (or smooth submanifolds M) in E may be computed in a universal way from a certain integral current, canonically associated to P (or M), living in the tangent sphere bundle SE = E x S~ (cf. [19], [20], [6]). If M is smooth, then this current is given by integration over the canonically oriented (n l)-manifold N(M) of unit normals to M. We may associate a similar object N(P) to a polyhedron P although N(P) is no longer a submanifold of SE, it is an integral current of dimension n 1, called the normal cycle to P. To obtain the curvature measures of P, we observe that there are universal differential (n l)-forms κ0, , κn_{ in SE n such*hat the curvature measures of P are given by