Comprehensive studies of cut locus

切割轨迹综合研究

基本信息

批准号：
17540085
负责人：
ITOH Jin-ichi
金额：
$ 2.3万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540085/
关键词：
geodesies cut locus 1'st conjugate locus Liouville manifold guadric surface distance function farthest point unfolding of polyhedron Liouvolle曲面単純閉擬測地線多面体の展閲 Liouville曲面モース理論

项目摘要

We studied the cut loci and related several topics, for example, the farthest points or simple closed geodesies on convex surfaces and got many results.As the problem to determine the cut locus, we proved that some compact Liouville manifolds have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds. We discussed on an extension to general dimension of Jacobi's last theorem(conjugate loci on ellipsoids have exact four cusps), and we got some remarkable progress and are planning to continue the research. Moreover, as the related topics, we got a modern proof of thread constructions of general quadric(hyper)surfaces by using the first integral (jointed work with K. Kiyohara).As the problem to study structures of cut locus, under some non-degenerating assumption we proved that the cut locus admits a nice stratification, some cone structure locally. Under stronger assumptions we have simpler procedure of Morse theory by using of critical points of distance functions (jointed work with T. Sakai).We established, for general convex surfaces, inequalities involving the diameter, the area and the length of simple closed quasi-geodesics (jointed work with C. Vilcu).Using the above simple closed quasi-geodesics we proved that any polyhedra are unfolded to a planar simple polygon by some cutting (jointed work with J. O'Rourke, C. Vilcu).We discussed several other unfolding by using simple clodes quasi geodesics, also

我们研究了剪切的基因座，并将几个主题与凸面上的最远点或简单的封闭测量线相关联，并获得了许多结果。当问题确定切割基因座时，我们证明了某些紧凑的liouville歧管具有一般点的剪切位点，即一般点的剪切位点是平稳嵌入了封闭的封闭式impormension immimensimensimensimensimensimensimensimensimensimensimensimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentimentiment的属性。具有不同轴的椭圆形是这种歧管的典型例子。我们讨论了雅各比的最后一个定理的一般维度的扩展（椭圆形的结合位点具有四个尖端），我们取得了一定的进步，并计划继续进行研究。此外，作为相关主题，我们通过使用第一个积分（与K. kiyohara的连接工作）获得了通用四边形（超级）表面的线程结构的现代证明。由于研究切割基因座的结构的问题，在某些非层次的假设下，我们证明，我们证明了cutus cutus cutus cuts cut supple cut supple cutse conse noce sentraciention a pene sole bene local inste ince ly Blocy。 Under stronger assumptions we have simpler procedure of Morse theory by using of critical points of distance functions (jointed work with T. Sakai).We established, for general convex surfaces, inequalities involving the diameter, the area and the length of simple closed quasi-geodesics (jointed work with C. Vilcu).Using the above simple closed quasi-geodesics we proved that any polyhedra are unfolded to a planar simple多边形通过一些切割（与J. O'Rourke，C。Vilcu的连接工作）。我们通过使用简单的Clodes quasi Geodesics讨论了其他几个展开