子流形的勒让德对偶及其奇点分类
发布时间：2018-07-17 20:52
【摘要】：本文以Legendrian对偶为主线,利用Legendrian奇点理论和Lagrangian奇点理论解决了一些具有Legendrian对偶关系的子流形的奇点分类问题.首先,我们不仅发现了伪欧氏空间中单参数族伪球之间的Legendrian对偶定理,还利用它和Legendrian奇点理论刻画了n维Anti de Sitter空间中的Lorentzian超曲面,指标为2的伪球中的Lorentzian超曲面及nullcone中的Lorentzian超曲面的外在微分几何性质.我们利用Legendrian对偶定理验证了上述子流形既存在nullcone高斯映射又存在φ-伪球高斯映射.其次,我们解决了指标为2的四维伪欧氏空间中的Lorentzian超曲面的Anti de Sitter高斯映射,以及三维de Sitter空间中的类空曲线的光锥对偶曲面和双曲对偶曲面的奇点分类问题.最后,我们利用三维欧氏空间中的球面Legendrian对偶,刻画了正则曲线的球面指标线之间的对偶关系,并应用相对平行标架场和奇点理论给出了Bishop球面指标线、Bishop球面Darboux像、Bishop对偶曲面及Bishop直纹曲面的奇点分类.本文共分为五章.第一章是引言.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.最后,简要阐述了全文的研究内容和结构安排.第二章主要介绍Legendrian奇点理论和Lagrangian奇点理论的一些基本概念和结论.第三章主要证明了伪欧氏空间中单参数族伪球之间的Legendrian对偶定理,并应用Legendrian对偶定理和Legendrian奇点理论研究了三种伪球上的Lorentzian超曲面的几何性质.对于四维伪欧氏空间中的Lorentzian超曲面,我们应用Lagrangian奇点理论解决了它们的Anti de Sitter高斯映射的奇点分类问题.第四章主要研究三维de Sitter空间中类空曲线的对偶曲面的奇点.三维de Sitter空间中的类空曲线存在两对对偶曲面.根据它们的位置向量所在的空间,分别称为第一光锥对偶曲面、第二光锥对偶曲面和第一双曲对偶曲面、第二双曲对偶曲面.我们证明了第一光锥对偶曲面和第一双曲对偶曲面是存在尖棱型和燕尾型奇点的曲面,而另外两个对偶曲面都是正则曲面.在Legendrian对偶定理的帮助下,我们揭示了类空曲线和这些曲面之间的对偶关系.通过对光锥高度函数和类时高度函数的研究,我们发现了刻画第一光锥对偶曲面和第一双曲对偶曲面的奇点的几何不变量.最后,我们给出了一个具体的例子.第五章主要利用三维欧氏空间中的球面Legendrian对偶刻画了正则曲线的球面指标线之间的对偶关系.依据相对平行标架场和奇点理论,解决了Bishop球面指标线、Bishop球面Darboux像、Bishop对偶曲面及Bishop直纹曲面的奇点分类问题.我们还得到了Bishop斜螺线的一些性质.最后,我们给出了两个具体例子.
[Abstract]:Taking Legendrian duality as the main line, this paper solves the singularity classification problems of some submanifolds with Legendrian duality by using Legendrian singularity theory and Lagrangian singular point theory. First of all, we not only find the Legendrian duality theorem between pseudo spheres of one parameter family in pseudo Euclidean space, but also use it and Legendrian singular point theory to characterize Lorentzian hypersurfaces in n-dimensional Anti de sitter space. The exterior differential geometric properties of Lorentzian hypersurfaces in pseudo sphere and nullcone hypersurface in nullcone. By using the Legendrian duality theorem, we prove that the above submanifolds have both nullcone Gao Si maps and 蠁 -pseudospherical Gao Si mappings. Secondly, we solve the Anti de sitter Gao Si mapping of Lorentzian hypersurfaces in 4-dimensional pseudo-Euclidean spaces with index 2, and the singularities classification of the optical cone dual surfaces and hyperbolic dual surfaces of space-like curves in three-dimensional de sitter space. Finally, using the spherical Legendrian duality in the three-dimensional Euclidean space, we characterize the duality between the spherical index lines of the regular curve. By using the relative parallel frame field and the singularity theory, the singularity classification of Bishop dual surface and Bishop straight surface of Bishop spherical index line and Bishop spherical Darboux image is given. This paper is divided into five chapters. The first chapter is the introduction. This paper mainly introduces the content, development and background of singularity theory application research. Finally, the research content and structure arrangement of the paper are briefly described. In the second chapter, some basic concepts and conclusions of Legendrian singular point theory and Lagrangian singular point theory are introduced. In chapter 3, we prove the Legendrian duality theorem between the family of pseudo-spheres in pseudo-Euclidean spaces, and apply Legendrian duality theorem and Legendrian singularity theory to study the geometric properties of Lorentzian hypersurfaces on three pseudo-spheres. For Lorentzian hypersurfaces in four dimensional pseudo Euclidean spaces, we apply Lagrangian singular point theory to solve the singularity classification problem of their Anti de sitter Gao Si maps. In chapter 4, we study the singularities of the dual surfaces of space-like curves in three-dimensional de Sitter space. There are two pairs of dual surfaces on the space-like curves in the three-dimensional de Sitter space. According to the space where their position vectors are located, they are called the first optical cone dual surface, the second optical cone dual surface, the first hyperbolic dual surface and the second hyperbolic dual surface. We prove that the first optical cone dual surface and the first hyperbolic dual surface are surfaces with sharp edges and swallowtail singularities, while the other two dual surfaces are regular surfaces. With the help of Legendrian's duality theorem, we reveal the duality relationship between space-like curves and these surfaces. By studying the height function of the optical cone and the time-like height function, we find the geometric invariants that characterize the singularities of the first optical cone dual surface and the first hyperbolic dual surface. Finally, we give a concrete example. In chapter 5, the duality of spherical index lines of regular curves is characterized by using the spherical Legendrian duality in three dimensional Euclidean space. Based on the relative parallel frame field and singularity theory, the problem of singularity classification of Bishop dual surface and Bishop straight line surface with Bishop spherical index line and Bishop spherical Darboux image is solved. We also obtain some properties of Bishop's oblique helix. Finally, we give two concrete 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】：2015
【分类号】：O186.1
[Abstract]:Taking Legendrian duality as the main line, this paper solves the singularity classification problems of some submanifolds with Legendrian duality by using Legendrian singularity theory and Lagrangian singular point theory. First of all, we not only find the Legendrian duality theorem between pseudo spheres of one parameter family in pseudo Euclidean space, but also use it and Legendrian singular point theory to characterize Lorentzian hypersurfaces in n-dimensional Anti de sitter space. The exterior differential geometric properties of Lorentzian hypersurfaces in pseudo sphere and nullcone hypersurface in nullcone. By using the Legendrian duality theorem, we prove that the above submanifolds have both nullcone Gao Si maps and 蠁 -pseudospherical Gao Si mappings. Secondly, we solve the Anti de sitter Gao Si mapping of Lorentzian hypersurfaces in 4-dimensional pseudo-Euclidean spaces with index 2, and the singularities classification of the optical cone dual surfaces and hyperbolic dual surfaces of space-like curves in three-dimensional de sitter space. Finally, using the spherical Legendrian duality in the three-dimensional Euclidean space, we characterize the duality between the spherical index lines of the regular curve. By using the relative parallel frame field and the singularity theory, the singularity classification of Bishop dual surface and Bishop straight surface of Bishop spherical index line and Bishop spherical Darboux image is given. This paper is divided into five chapters. The first chapter is the introduction. This paper mainly introduces the content, development and background of singularity theory application research. Finally, the research content and structure arrangement of the paper are briefly described. In the second chapter, some basic concepts and conclusions of Legendrian singular point theory and Lagrangian singular point theory are introduced. In chapter 3, we prove the Legendrian duality theorem between the family of pseudo-spheres in pseudo-Euclidean spaces, and apply Legendrian duality theorem and Legendrian singularity theory to study the geometric properties of Lorentzian hypersurfaces on three pseudo-spheres. For Lorentzian hypersurfaces in four dimensional pseudo Euclidean spaces, we apply Lagrangian singular point theory to solve the singularity classification problem of their Anti de sitter Gao Si maps. In chapter 4, we study the singularities of the dual surfaces of space-like curves in three-dimensional de Sitter space. There are two pairs of dual surfaces on the space-like curves in the three-dimensional de Sitter space. According to the space where their position vectors are located, they are called the first optical cone dual surface, the second optical cone dual surface, the first hyperbolic dual surface and the second hyperbolic dual surface. We prove that the first optical cone dual surface and the first hyperbolic dual surface are surfaces with sharp edges and swallowtail singularities, while the other two dual surfaces are regular surfaces. With the help of Legendrian's duality theorem, we reveal the duality relationship between space-like curves and these surfaces. By studying the height function of the optical cone and the time-like height function, we find the geometric invariants that characterize the singularities of the first optical cone dual surface and the first hyperbolic dual surface. Finally, we give a concrete example. In chapter 5, the duality of spherical index lines of regular curves is characterized by using the spherical Legendrian duality in three dimensional Euclidean space. Based on the relative parallel frame field and singularity theory, the problem of singularity classification of Bishop dual surface and Bishop straight line surface with Bishop spherical index line and Bishop spherical Darboux image is solved. We also obtain some properties of Bishop's oblique helix. Finally, we give two concrete 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】：2015
【分类号】：O186.1
【相似文献】
相关期刊论文 前10条
1 杨清华;一类三次系统的奇点量问题[J];韶关学院学报(自然科学版);2001年09期
2 刘锐宽;雅可比型系统奇点稳定性的判断[J];辽宁工程技术大学学报(自然科学版);2002年06期
3 潘雪军;章丽娜;;一类拟五次系统的奇点量与中心条件[J];丽水学院学报;2008年05期
4 王伟;;一类二元边界奇点的识别[J];塔里木大学学报;2009年01期
5 杨云锋;夏小刚;冯卫兵;;奇点定义注记[J];五邑大学学报(自然科学版);2010年03期
6 郭瑞芝;谢瑞芳;;三元边界奇点在R~*_H-等价下的分类及识别[J];湖南农业大学学报(自然科学版);2010年05期
7 吕百达;;奇点光学概论[J];四川师范大学学报(自然科学版);2011年06期
8 张川;;几类非线性系统奇点的分类及相图[J];旅游纵览(行业版);2012年06期
9 刘世泽;;关于奇点分类的问题[J];内蒙古大学学报(自然科学版);1964年02期
10 李存富;;平面三次系统无z镀娴惴治，
本文编号：2130898
本文链接：https://www.wllwen.com/shoufeilunwen/jckxbs/2130898.html
