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The structure of cut locus and global Riemannian geometry

割轨迹的结构与全局黎曼几何

基本信息

批准号：
09440037
负责人：
ITOH Jin-ichi
金额：
$ 3.97万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440037/
关键词：
cut locus distance function Riemannian manifold Hausdorff measure Lipschitz continuous Voronoi domain geodesics polytope ハウスドルフ測度 lipsehitz性

项目摘要

Recently, it was showed that the total length of cut locus of Riemannian surface is finite in the study of the structure of cut locus and by using this Ambrose's problem of surface was answered affirmatively. The purposes of this research are to study the structure of cut locus of Riemannian manifold and to study global Riemannian geometry by using the above results.The most main result that the distance function to the cut locus is Lipshitz continuous is proved the 1'st year, and now printing (joint work with M.Tanaka). It follows that the distance is derived naturally on the cut locus. It happens the following interesting question "what is the natural geometric structure on the cut locus?''In the study of some kind of stratification of cut locus of CィイD1∞ィエD1 Riemannian manifold, at the beginning, we tried the problem "Is the cut locus locally a submanifold around any cut point except for any subset which is of local dimensional Hausdorff measure zero?" We proved this affirmatively by very complicated method (j.w. with M.Tanaka.). Now, we are searching simpler method.Also we studied the set of critical points of distance function which is closely related with the cut locus. We proved last year that the set of all critical values of the distance function for a submanifold of a 3-dimensional complete Riemannian manifold is of Lebesgue measure zero (Sard type theorem). Now, we extend this result that ィイD71(/)2ィエD7-dimensional Hausdorff measure is zero (j.w. with M.Tanaka). By using method we expect that in the 4-dimensional case it is of Lebesgue measure zero.We cannot answered Ambrose's problem of general dimension, but we get the evaluation of face of Voronoi domain in any Hadamard manifold and showed that the subset (essential cut locus) of the cut locus which is contained any critical point of distance function is not so complicated in the case of low dimensional convex polytope.

最近，在研究切割轨迹结构的过程中，证明了黎曼曲面切割轨迹的总长度是有限的，并由此肯定地回答了曲面的Ambrose问题。本研究的目的是研究黎曼流形切割轨迹的结构，并利用上述结果来研究全局黎曼几何。最主要的结果是到切割轨迹的距离函数是利普希兹连续的，这是在1年被证明的，现在正在印刷（与田中先生联合工作）。由此可见，距离是在切割轨迹上自然推导出来的。这就引出了一个有趣的问题"切割轨迹上的自然几何结构是什么？”削减一些分层的研究轨迹CィイD1∞ィエD1黎曼流形,在一开始,我们尝试的问题”是减少轨迹周围局部子流形切割点除了任何子集的当地维豪斯道夫测度零?”我们用非常复杂的方法肯定地证明了这一点（j.w.与田中）。现在，我们正在寻找更简单的方法。研究了与切割轨迹密切相关的距离函数的临界点集合。去年我们证明了三维完全黎曼流形的子流形的距离函数的所有临界值的集合是勒贝格测度为零的（Sard型定理）。现在，我们推广了这一结果，即j.w. with M.Tanaka证明了j.w. d . 7维Hausdorff测度为零。通过使用该方法，我们期望在四维情况下它是勒贝格测度零。我们不能回答一般维数的Ambrose问题，但我们得到了任意Hadamard流形上Voronoi区域面的评价，并证明了在低维凸多面体的情况下，包含距离函数任意临界点的切割轨迹的子集（本质切割轨迹）并不复杂。