非平坦空间中子流形的微分几何

发布时间：2017-12-31 02:37

本文关键词：非平坦空间中子流形的微分几何　出处：《东北师范大学》2017年博士论文　论文类型：学位论文

【摘要】：本文主要研究了非平坦空间中子流形的一些微分几何性质,并且利用Lagrangian奇点理论和Legendrian奇点理论对一些子流形的奇点进行了分类.2003年,B.Y.Chen提出了从切曲线的概念[23],这类曲线的特殊性类似于平面曲线和球面曲线的特殊性.我们在非平坦空间中推广了从切曲线的概念,着重研究了类光从切曲线的一些几何性质.同时,本文利用奇点理论解决了两类常曲率空间中子流形的奇点分类问题,其中这两类空间分别是有着正的常截面曲率的de Sitter空间和负的常截面曲率的anti de Sitter空间.我们探讨了de Sitter空间中的类时曲线和anti de Sitter空间中的类空曲线,研究了它们的法超曲面,焦曲面,渐屈线和单参数类光超曲面的性质,揭示了这些子流形的奇点与几何不变量之间的关系.最后,本文考虑了一般类光曲面上曲线的三种伪球法达布像.从奇点的角度,给出了伪球法达布像的奇点分类;从切触几何的角度,揭示了曲线与曲面切片曲线在奇点处的切触关系;从Legendrian对偶的角度,探索了伪球法达布像与达布标架场之间的对偶关系.本文共分为七章.第一章是引言部分,主要介绍了本文的背景,奇点理论发展概况和应用研究的内容,并简要阐述了全文的研究内容和结构安排.第二章主要介绍了半欧氏空间中的一些非平坦子空间和子流形的基本概念.第三章主要介绍了非平坦空间中从切曲线的定义,着重研究了类光从切曲线的几何性质.第四章主要研究了三维de Sitter空间中类时曲线的法超曲面的奇点分类.第五章主要研究了de Sitter空间中类时Sabban曲线的渐屈线和焦曲面的几何性质.第六章主要研究了三维anti de Sitter空间中类空曲线的单参数类光超曲面的奇点分类,并且我们给出了具体例子.第七章主要研究了类光曲面上曲线的三种伪球法达布像,从Legendrian对偶的角度,讨论了伪球法达布像与达布标架场之间的关系,最后我们给出了具体例子。
[Abstract]:This paper mainly studies some differential geometry of Submanifolds in non flat space, and by using the singularity Lagrangian singularity theory and Legendrian singularity theory for some submanifolds are classified in.2003, B.Y.Chen proposed the concept of [23] from the tangent curve, the curve of particularity is similar to that of a plane curve and spherical curve. We in the non flat space extends from the concept of cutting curve, focusing on the kind of light from some geometric properties of cutting curves. At the same time, the singularity theory to solve the classification problem of two types of singularities of Submanifolds in constant curvature space, wherein the two space are a constant positive sectional curvature de Sitter negative space and constant sectional curvature anti de Sitter space. We discussed the spacelike curve de in Sitter space and anti de timelike curve in Sitter space, and studied the method of super Curved focal surface, evolute and single parameter properties of light hypersurface, reveals the relationship between the singularity and geometric invariants of these submanifolds. Finally, this paper considers three kinds of general Fadabu pseudo spherical light curves on the surface. Like from the singular angles, given the singularities of pseudo spherical Darboux method as the contact geometry; from the point of view, reveals the relationship between the contact curve and surface slice curves at singular point; Legendrian from dual perspective, explores the ball like Fadabu pseudo dual relationship between the subject and the Darboux frame field. This paper is divided into seven chapters. The first chapter is the introduction part, mainly introduced in this paper, the background, development of content singularity theory and applied research, and briefly introduces the research contents and the structure of the dissertation. The second chapter mainly introduces some non flat subspace and the basic concept of semi Submanifolds in Euclidean space. The third chapter The definition of tangent curve from non flat space, focusing on the kind of light from the geometric properties of tangent curve. The fourth chapter mainly studies the classification of singularities of 3D de timelike curve in Sitter space method hypersurface. The fifth chapter mainly studies the class of Sabban de curve in Sitter space evolute lines and geometric properties of coke the single parameter surface. The sixth chapter mainly studies the 3D anti de Sitter space in a spacelike curve light hypersurface singularities, and we give specific examples. Three kinds of pseudo spherical Fadabu seventh chapter mainly studies the light curves on the surface like, from Legendrian dual perspective, discussed the pseudo spherical Fadabu the relationship between the field as standard frame and Darboux, finally we give the specific examples.

【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】：O186.1

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