Integrable geodesic flows and related problems

可积测地流及相关问题

基本信息

批准号：
16540069
负责人：
KIYOHARA Kazuyoshi
金额：
$ 2.24万
依托单位：
Okayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540069/
关键词：
ellipsoid cut locus conjugate locus Liouville manifold Jacobi Integrable geodesic flow tri-axial ellipsoid quadratic surface 共役蹠リウヴィル曲面極射影同値双曲面

项目摘要

We made a series of researches on "cut locus". First, we showed that the cut locus of any point on two-dimensional ellipsoids which is not an umbilic point is a segment of the curvature line passing through the antipodal point. Moreover, we proved that the conjugate locus of that point has exactly four cusps, and they appear on the two curvature lines passing through the antipodal point. The latter result was stated in Jacobi's "Lectures on dynamical systems" in the case of rotational ellipsoids, and had remained unproved.Secondly, we showed that on certain Liouville surfaces including the ellipsoids the cut locus of a general point is "simple", i.e., a curve segment in compact case, and either empty or a curve segment or two curve segments in noncompact case. In particular, in the case of (a connected component of) two-sheeted hyperboloids, it was proved that there are two cases : In one case all of the above three types of cut loci appear ; and in the other case only connected cut loci appear. Thirdly, we proved that for a certain class of Liouville manifold diffeomorphic to the sphere, the cut locus of a general point is diffeomorphic to the closed disk of codimension one.. In particular, this class contains the ellipsoids whose principal axes have distinct length.Also, we studied "Hermite-Liouville manifolds", which are not necessarily Kaehler-Liouville manifolds, and completely determined their local structures. Among them are involved the cases where the infintesimal automorphisms are not associated. Moreover, we constructed Hermite-Liouville manifolds over the complex projective space as a complexification of real Liouville manifolds over the real projective space. Our construction involves the parameters which almost meets the local possibility.

我们对“切割轨迹”进行了一系列的研究。首先，我们证明了二维椭球上任意非脐点的点的切割轨迹是通过对映点的曲率线的一段。此外，我们证明了该点的共轭轨迹恰好有四个尖点，并且它们出现在通过对映点的两条曲率线上。后一个结果在雅可比的“动力系统讲座”中针对旋转椭球体进行了阐述，但尚未得到证实。其次，我们证明在某些包括椭球体的刘维尔曲面上，一般点的切割轨迹是“简单的”，即，紧凑情况下的一条曲线段，以及非紧凑情况下的空曲线段或曲线段或两个曲线段。特别地，在二片双曲面（的连通分量）的情况下，证明了存在两种情况：在一种情况下，上述三种类型的切割轨迹都出现；在一种情况下，出现了所有上述三种类型的切割轨迹。在另一种情况下，仅出现连接的切割基因座。第三，我们证明了对于一类与球体微分同胚的刘维尔流形，一般点的切割轨迹与余维一闭圆盘微分同胚。特别地，该类包含主轴具有不同长度的椭球体。此外，我们还研究了“Hermite-Liouville流形”，它不一定是Kaehler-Liouville流形，并完全确定了它们的局部结构。其中涉及无穷小自同构不相关的情况。此外，我们在复射影空间上构建了埃尔米特-刘维尔流形，作为实射影空间上的实刘维尔流形的复化。我们的施工涉及的参数几乎满足当地的可能性。

项目成果

期刊论文数量（28）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Gauss-type curvatures and tubes for polyhedral surfaces

多面体表面的高斯型曲率和管

DOI：
发表时间：
2005
期刊：
Kumamoto J. Math 18
影响因子：
0
作者：
M.Arkovitz;H.Oshima;J.Strom;J. Itoh and F. Ohtsuka;F. Ohtsuka;J. Itoh and T. Zamfirescu;J. Itoh
通讯作者：
J. Itoh

Appendix to "Some metric invariants of spheres and Alexandrov spaces II"

附录“球体和亚历山德罗夫空间 II 的一些度量不变量”

DOI：
发表时间：
2005
期刊：
Math.J.Okayama Univ. 4
影响因子：
0
作者：
Y. Mori;R. Sawae;M. Kawamura;T. Sakata;K. Takarabe;K.Kiyohara
通讯作者：
K.Kiyohara

Simplicies passing through a hole

简单地穿过一个洞

DOI：
发表时间：
2005
期刊：
J. of Geometry 83
影响因子：
0
作者：
M.Arkovitz;H.Oshima;J.Strom;J. Itoh and F. Ohtsuka;F. Ohtsuka;J. Itoh and T. Zamfirescu;J. Itoh;J. Itoh and T. Zamfirescu
通讯作者：
J. Itoh and T. Zamfirescu

Appendix to "Some metric invariants of Sphere and Alexandrov spaces II"

附录“球体和亚历山德罗夫空间 II 的一些度量不变量”