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“切割基因座”进行了一系列研究。首先，我们证明了二维椭圆形上任何点的切割基因座，这不是脐带点是穿过抗焦点点的曲率线的一部分。此外，我们证明了该点的共轭基因座完全具有四个尖端，并且它们出现在穿过抗肌点的两个曲率线上。在旋转椭圆形的情况下，在雅各比的“动力学系统的讲座”中说明了后一个结果，并且一直没有得到证实。特别是，对于（连接的）两片双粘性的组件，证明有两种情况：在一种情况下，出现了上述三种切割基因座的所有类型；在另一种情况下，仅出现连接的切割基因座。第三，我们证明，对于一定类别的一定类别的liouville多种形态，一般点的切割基因座对编成粘合一体的封闭磁盘是不同的。尤其是，此类级别包含其原理轴具有独特的长度的椭圆形的，我们完全不一定地研究了“ hermite-liouville sully and kae fivelly kee fistry strimational kae firly soilly striply siplimand kae firoleds”，“ kae firly是”，这是“ kee firolly sullimandimate kae firoleds”。结构。其中涉及无义自动形态不相关的情况。此外，我们在复杂的投影空间上构建了Hermite-liouville歧管，作为对真实的liouville歧管在真实的投射空间上的复杂化。我们的构建涉及几乎符合当地可能性的参数。

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”