光滑曲线和曲面的微分几何

发布时间：2018-05-30 08:18

本文选题：勒让德浸入 + 渐屈线　；参考：《东北师范大学》2016年博士论文

【摘要】：本文主要研究了半欧氏空间中的光滑曲线和光滑曲面在奇点邻近的微分几何.2009年,几何学家Saji, Umehara, Yamada在美国数学年刊发表的文章中系统的阐述了曲面在尖楞处的曲率函数的定义并且解释了高斯曲率在尖楞和燕尾处的特征.这篇文章是研究子流形在奇点邻近微分几何性质的一个里程碑.在这个时期,许多数学工作者投入到了子流形在奇点邻近几何性质的研究当中.本文首先关注了双曲平面上带有奇点的光滑曲线,给出了奇点邻近曲率和渐屈线的概念,描述了三种伪球上的渐屈线的不同特征,进而研究了曲线的奇点和测地顶点之间的关系.其次,本文考虑了指标为二的四维半欧氏空间中的伪类光曲线和偏类光曲线,研究了它们的光锥高斯曲面的性质,揭示了它们的类光超曲面的奇点与一些几何不变量之间的关系.最后,本文从类光几何的角度考虑了四维Anti de Sitter空间中类空曲面的拐点和H奇点,解决了拐点的分类和识别问题,探索了拐点和H奇点的关系。本文共分为四章.第一章引言,主要介绍奇点理论应用研究的内容,发展概况和本文的背景,并简要阐述了全文的研究内容和结构安排。第二章主要介绍了光滑曲线和光滑曲面相关的子流形的微分几何和奇点理论的一些基本概念和结论。第三章主要研究了光滑曲线及其生成的子流形在奇点邻近的微分几何.简要介绍了欧氏平面上奇异曲线的几何性质,进而研究了双曲平面上的奇异曲线.在奇点处定义了曲率的概念,并进一步研究了多重渐屈线和四顶点定理.对于指标为二的四维半欧氏空间中伪类光曲线和偏类光曲线,我们应用Legendrian奇点理论解决了它们的类光超曲面的奇点分类问题。第四章主要研究了四维Anti de Sitter空间中类空曲面的拐点的识别问题.我们知道曲面上每一点都对应着一个曲率椭圆,当曲率椭圆退化成径向线段时,对应的点被称为拐点.依赖于退化的曲率椭圆与拐点的位置关系,我们可以将拐点分为三类,即实型拐点、虚型拐点、平坦型拐点.首先,我们用传统的办法给出了判断拐点类别的方法.其次,我们从类光几何的角度分别揭示了实型拐点、虚型拐点、平坦型拐点的等价条件.最后,我们给出平均方向曲线的微分方程,并且指出H奇点是由拐点和稳定点构成的。
[Abstract]:In this paper, we study the differential geometry of smooth curves and smooth surfaces near singularities in semi-Euclidean spaces. In an article published in the American Journal of Mathematics, the geometric scientist Saji, Umehara, Yamada has systematically expounded the definition of curvature function at the tip of the surface and explained the characteristics of the curvature of the Gao Si at the tip of the corrugated and the swallow-tail. This paper is a milestone in the study of differential geometry of submanifolds near singularities. During this period, many mathematical workers devoted themselves to the study of the geometric properties of submanifolds near singularities. In this paper, we first focus on the smooth curves with singularities on the hyperbolic plane, give the concepts of the adjacent curvature and the involute line of the singularities, and describe the different characteristics of the evolutional lines on the three pseudo spheres. Furthermore, the relationship between the singularity of the curve and the geodesic vertex is studied. Secondly, in this paper, we consider the pseudo-photoluminescence curve and the biased photophore curve in the four-dimensional semi-Euclidean space with index two, and study the properties of their optical cone Gao Si surfaces. The relationship between the singularity of their photonic hypersurfaces and some geometric invariants is revealed. Finally, in this paper, the inflection points and H singularities of space-like surfaces in four-dimensional Anti de Sitter spaces are considered from the point of view of light-like geometry. The problem of classification and recognition of inflection points is solved, and the relationship between inflection points and H singularities is explored. This paper is divided into four chapters. The first chapter introduces the content, development and background of the application of singularity theory, and briefly describes the research content and structure of this paper. In chapter 2, some basic concepts and conclusions of differential geometry and singularity theory of submanifolds related to smooth curves and smooth surfaces are introduced. In chapter 3, we study the differential geometry of smooth curves and their generated submanifolds near singularities. The geometric properties of singular curves on Euclidean plane are briefly introduced, and the singular curves on hyperbolic plane are studied. The concept of curvature at singularities is defined, and the multiple involute and four-vertex theorems are further studied. For pseudo-photophore curves and partial optical-like curves in four-dimensional semi-Euclidean space with index two, we apply Legendrian singularity theory to solve the singularity classification problem of their photohypersurfaces. In chapter 4, the problem of recognizing inflection points of space-like surfaces in four dimensional Anti de Sitter spaces is studied. We know that every point on the surface corresponds to an ellipse of curvature, and when the ellipse of curvature degenerates into a radial segment, the corresponding point is called the inflection point. Depending on the position relationship between the degenerate curvature ellipse and the inflection point, we can divide the inflection point into three categories, that is, the real inflection point, the virtual inflection point and the flat inflection point. First of all, we use the traditional method to determine the inflection point category. Secondly, we reveal the equivalent conditions of real type inflection point, virtual type inflection point and flat type inflection point from the point of view of similar optical geometry. Finally, we give the differential equation of the mean direction curve, and point out that the singularity of H is composed of the inflection point and the stable point.
【学位授予单位】：东北师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O186.1

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