3维Anti de Sitter空间中的特殊曲线和特殊直纹面

发布时间：2018-03-01 17:28

本文关键词： 3维Anti de Sitter空间第一类空贝特朗曲线类时贝特朗曲线主法测地面 null Cartan 曲线　出处：《东北师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：Anti de Sitter空间(简记为Ads-空间)是截面曲率为-1的Lorentzian空间型,在物理学中,Ads-空间是爱因斯坦方程的真空解之一.研究Ads-空间中的曲面和曲线既有具体的物理背景,又有几何学上的理论意义.本文主要研究3维Ads-空间中的特殊曲线和特殊直纹面的几何性质.第一章主要介绍了 3维Ads-空间中曲线的基本概念和基本定理.第二章给出了 3维Ads-空间中由第一类空正则曲线的主法测地线生成的主法测地面的定义及其性质,介绍了 3维Ads-空间中第一类空贝特朗曲线的定义及其相关性质,并给出了它与第一类空正则曲线的测地面的关系.第三章给出了 3维Ads-空间中由类时正则曲线的主法测地线生成的主法测地面的定义及其性质,介绍了3维Ads-空间中类时贝特朗曲线的定义及其相关性质,并给出了它与类时正则曲线的测地面的关系.第四章介绍了3维Ads-空间中null Cartan曲线以及与之相关的null曲面的相关定义,给出了null Cartan曲线的副法高度函数的定义并介绍了两个新的几何不变量,利用奇点理论中关于函数的开折理论以及这些不变量给出了null曲面奇点的分类,并给出了奇点的几何性质.
[Abstract]:Anti de Sitter (abbreviated as Ads- space) is the sectional curvature space for the Lorentzian -1, in physics, Ads- space is a vacuum solution of the Einstein equation. Ads- space curved surface and curve in both the physical background of concrete, and the geometry theory significance. This paper mainly studies 3 special curves dimensional Ads- space and special geometric properties of the ruled surface. The first chapter mainly introduces the basic concept of 3 dimensional curve in Ads- space and the basic theorem. The second chapter gives the definition and properties of geodesic spacelike curve generated by the first canonical method of 3 dimensional Ads- space method of ground measurement, introduces the definition of 3 dimensional Ads- space first spacelike Bertrand curve and its related properties, and gives the relationship between the measured ground with the first empty regular curve. The third chapter gives 3 dimensional Ads- space by the main class of regular curves The definition and character of method of generating geodesic method of ground measurement, introduces the definition of timelike Bertrand curves of 3 dimensional Ads- spaces and some properties, and gives the relationship between it and the class of regular ground measurement curve. The fourth chapter introduces the null Cartan curve of 3 dimensional Ads- spaces and related definitions related to null surface the definition of null is given pair of height function Cartan curve and introduced two new geometric invariants, the unfolding theory about the function in singularity theory and presents a classification of these invariants null surface singularities, and gives the geometric properties of the singularity.

【学位授予单位】：东北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.1

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