四元数双曲空间等距球与不动点的关系

发布时间：2018-05-28 13:09

本文选题：四元数 + 双曲空间　；参考：《五邑大学》2015年硕士论文

【摘要】：本文基于Mobius变换等距球的性质和作用,得到了四元数Mobius变换g的一个分解g=tfO。利用旋转变换O与对应Mobius变换g的不动点的关系,得到一定条件下四元数Mobius变换的分类。双曲等距映射可分为斜驶元素、抛物元素和椭圆元素三类。缘于其不动点集的多样性,椭圆元素是三类元素中最为特殊的一类,高维双曲轨道流形的性质一般都与此相关。本文将分别得到低维实双曲空间,复双曲空间和四元数双曲空间的椭圆元素的标准形及他们的不动点集的相关性质。本文安排如下:第一章我们介绍了课题的背景知识及我们得到的主要结果内容;第二章主要介绍了本文所要用到的四元数和四元数矩阵基本知识;第三章主要介绍了四元数Mobius变换的一些基本知识,得到了四元数Mobius变换g的一个分解g=tfO,并证明了本文的主要结论;第四章分别讨论了低维实双曲空间,复双曲空间和四元数双曲空间的椭圆元素不动点集的相关情况。给出了四元数双曲空间椭圆元素在Siegel域模型下的一种分类。
[Abstract]:In this paper, based on the properties and actions of the equidistant sphere of Mobius transform, we obtain a decomposition of the quaternion Mobius transform g. By using the relation between the rotation transformation O and the fixed points of the corresponding Mobius transform g, the classification of the quaternion Mobius transform is obtained under certain conditions. Hyperbolic isometric mapping can be divided into three types: oblique elements, parabolic elements and elliptic elements. Because of the diversity of its fixed point sets, elliptic elements are the most special of the three kinds of elements, and the properties of high-dimensional hyperbolic orbital manifolds are generally related to them. In this paper, we obtain the canonical forms of elliptic elements in low dimensional real hyperbolic spaces, complex hyperbolic spaces and quaternion hyperbolic spaces and the related properties of their fixed point sets. The main contents of this paper are as follows: in chapter one, we introduce the background knowledge of the subject and the main results we get, the second chapter mainly introduces the basic knowledge of quaternion and quaternion matrix used in this paper. In chapter 3, we introduce some basic knowledge of quaternion Mobius transform, obtain a decomposition of quaternion Mobius transform g, and prove the main conclusions of this paper. Correlation of fixed point sets of elliptic elements in complex hyperbolic space and quaternion hyperbolic space. A classification of elliptic elements in quaternion hyperbolic space under Siegel domain model is given.
【学位授予单位】：五邑大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O181

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