3维空间中零Cartan曲线的零曲面及标架曲线的渐屈线

发布时间：2018-02-25 22:06

本文关键词： 三维Anti-de Sitter空间零Cartan曲线直纹零曲面类光子流形标架曲线渐屈线　出处：《东北师范大学》2017年博士论文　论文类型：学位论文

【摘要】：本文主要研究了指标为2的四维半欧氏空间的子流形H31中的零Cartan曲线的微分几何问题.首先根据零Cartan曲线本身的特点,建立Cartan Frenet标架,得到特殊的Cartan Frenet方程.然后定义与零Cartan曲线相关联的直纹零曲面及副法指标高度函数,揭示了在洛仑兹群作用下的零Cartan曲线的几何不变量和直纹零曲面的奇点的关系,应用Bruce等的奇点分类方法实现直纹零曲面的奇点分类.此外,本文还给出了三维欧氏空间中带有奇点的曲线的渐屈线定义,进而研究了它的性质,指出该定义可作为正则曲线渐屈线定义的推广.本文共分为四章.第一章引言,主要介绍奇点理论的发展概况,并简要阐述了本文的研究内容和基本框架.在第二章中,我们给出了指标为2的半欧式空间R_2~4中相关的预备知识.介绍了R_2~4空间的伪内积,伪向量积,伪正交,伪弧长参数,类空,类光及类时向量,类空曲线,类光曲线(零曲线),类时曲线等基本概念,并且给出了三维Anti-de Sitter空间,指标为2的三维de Sitter空间及三维开光锥等重要子流形的概念.特别地,我们在本章最后介绍了一个重要结论[29],该结论能够保证零曲线具有唯一的与切向量配对的零向量.第三章主要介绍三维Anti-de Sitter空间中零Cartan曲线的Cartan Frenet标架及Cartan Frenet方程.以此作为基本研究工具,运用Bruce的奇点分类方法,给出了由三维Anti-de Sitter空间中零Cartan曲线的主法指标线作为基线生成的直纹零曲面的奇点分类.具体分类结果见定理3.6.1.在第四章中,我们研究带有奇点的曲线的渐屈线的性质.首先我们给出了标架曲线的概念,并且对于标架曲线定义了一个移动适应标架.利用移动适应标架,我们定义了一些光滑函数,这些光滑函数对于研究标架曲线所起的作用类似于正则曲线的曲率.我们称这些函数为标架曲率.在此基础上,我们给出三维欧氏空间中标架曲线的渐屈线的定义.具体定义形式见定理4.2.1.值得一提的是,这个关于渐屈线的新定义与当曲线为正则曲线时渐屈线的经典定义是一致的.
[Abstract]:In this paper, the differential geometry of zero Cartan curve in the submanifold H31 of four-dimensional semi-Euclidean space with index 2 is studied. Firstly, according to the characteristics of zero Cartan curve, the Cartan Frenet frame is established. The special Cartan Frenet equation is obtained, and then the relations between the geometric invariant of the zero Cartan curve and the singularity of the straight zero surface are revealed by defining the straight-line zero surface associated with the zero Cartan curve and the index height function of the pair method, and the relation between the geometric invariants of the zero Cartan curve and the singularity of the straight zero surface under the Lorentz group is revealed. The singularity classification method proposed by Bruce et al is used to classify the singularities of straight zero surfaces. In addition, this paper gives the definition of the curve with singular points in three dimensional Euclidean space, and then studies its properties. It is pointed out that this definition can be used as a generalization of the definition of regular curve involution line. This paper is divided into four chapters. Chapter one introduces the development of singular point theory, and briefly describes the research content and basic framework of this paper. In this paper, we give the preparatory knowledge of the semi-European space R2K4 with index 2, and introduce the pseudo-inner product, pseudo-vector product, pseudo-orthogonal, pseudo-arc length parameter, space-like space, optical-like vector and time-like vector, space-like curve of R2C4 space. The basic concepts of photo-like curve (zero curve, time-like curve, etc.) are given. The concepts of three dimensional Anti-de Sitter space, three dimensional de Sitter space with index 2 and three dimensional open cone are given. In particular, some important submanifolds, such as 3 D Anti-de Sitter space, 3 D de Sitter space and 3 D open cone, are given. At the end of this chapter, we introduce an important conclusion [29], which can guarantee that zero curve has unique zero vector matched with tangent vector. Chapter 3 mainly introduces the Cartan Frenet frame and Cartan Frenet side of zero Cartan curve in three-dimensional Anti-de Sitter space. As a basic research tool, By using Bruce's singularity classification method, the singularity classification of straight zero surface generated by the main index line of zero Cartan curve in 3D Anti-de Sitter space is given. The classification results are shown in Theorem 3.6.1. In Chapter 4th, We study the properties of curves with singularities. Firstly, we give the concept of frame curves, and define a moving adaptive frame for frame curves. By using moving adaptive frames, we define some smooth functions. These smooth functions play a similar role to the curvature of regular curves in studying frame curves. We call these functions frame curvature. In this paper, we give the definition of the progressive curve of the frame curve in three dimensional Euclidean space. The form of the definition is shown in Theorem 4.2.1. It is worth mentioning that this new definition of the curve is consistent with the classical definition of the progressive curve when the curve is a regular curve.
【学位授予单位】：东北师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O186.11

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