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，结果表明，在研究切割基因座的结构的研究中，黎曼表面的切割基因座的总长度是有限的，并且通过使用该安布罗斯的表面问题得到了合理的回答。这项研究的目的是研究Riemannian歧管的切割基因座的结构，并通过使用上述结果研究全球Riemannian几何形状。最主要的结果是，切割基因座的距离函数是lipshitz Ryne的连续证明是1年级的，现在是1'，现在是印刷（与M.Tanaka的联合工作）。因此，距离自然派生在切割基因座上。发生了以下有趣的问题：“切割基因座上的自然几何结构是什么？”在研究某种类型的ciyd1∞d1riemannian歧管的切割基因座的某种分层时，一开始，我们尝试了问题。 M.Tanaka。）。 D7二维的Hausdorff测量为零（J.W.使用M.Tanaka）。在低维凸多属的情况下，功能并不是那么复杂。