子流形的勒讓德對偶及其奇點(diǎn)分類
發(fā)布時(shí)間：2018-07-17 20:52
【摘要】：本文以Legendrian對偶為主線,利用Legendrian奇點(diǎn)理論和Lagrangian奇點(diǎn)理論解決了一些具有Legendrian對偶關(guān)系的子流形的奇點(diǎn)分類問題.首先,我們不僅發(fā)現(xiàn)了偽歐氏空間中單參數(shù)族偽球之間的Legendrian對偶定理,還利用它和Legendrian奇點(diǎn)理論刻畫了n維Anti de Sitter空間中的Lorentzian超曲面,指標(biāo)為2的偽球中的Lorentzian超曲面及nullcone中的Lorentzian超曲面的外在微分幾何性質(zhì).我們利用Legendrian對偶定理驗(yàn)證了上述子流形既存在nullcone高斯映射又存在φ-偽球高斯映射.其次,我們解決了指標(biāo)為2的四維偽歐氏空間中的Lorentzian超曲面的Anti de Sitter高斯映射,以及三維de Sitter空間中的類空曲線的光錐對偶曲面和雙曲對偶曲面的奇點(diǎn)分類問題.最后,我們利用三維歐氏空間中的球面Legendrian對偶,刻畫了正則曲線的球面指標(biāo)線之間的對偶關(guān)系,并應(yīng)用相對平行標(biāo)架場和奇點(diǎn)理論給出了Bishop球面指標(biāo)線、Bishop球面Darboux像、Bishop對偶曲面及Bishop直紋曲面的奇點(diǎn)分類.本文共分為五章.第一章是引言.主要介紹奇點(diǎn)理論應(yīng)用研究的內(nèi)容,發(fā)展概況和本文的背景.最后,簡要闡述了全文的研究內(nèi)容和結(jié)構(gòu)安排.第二章主要介紹Legendrian奇點(diǎn)理論和Lagrangian奇點(diǎn)理論的一些基本概念和結(jié)論.第三章主要證明了偽歐氏空間中單參數(shù)族偽球之間的Legendrian對偶定理,并應(yīng)用Legendrian對偶定理和Legendrian奇點(diǎn)理論研究了三種偽球上的Lorentzian超曲面的幾何性質(zhì).對于四維偽歐氏空間中的Lorentzian超曲面,我們應(yīng)用Lagrangian奇點(diǎn)理論解決了它們的Anti de Sitter高斯映射的奇點(diǎn)分類問題.第四章主要研究三維de Sitter空間中類空曲線的對偶曲面的奇點(diǎn).三維de Sitter空間中的類空曲線存在兩對對偶曲面.根據(jù)它們的位置向量所在的空間,分別稱為第一光錐對偶曲面、第二光錐對偶曲面和第一雙曲對偶曲面、第二雙曲對偶曲面.我們證明了第一光錐對偶曲面和第一雙曲對偶曲面是存在尖棱型和燕尾型奇點(diǎn)的曲面,而另外兩個(gè)對偶曲面都是正則曲面.在Legendrian對偶定理的幫助下,我們揭示了類空曲線和這些曲面之間的對偶關(guān)系.通過對光錐高度函數(shù)和類時(shí)高度函數(shù)的研究,我們發(fā)現(xiàn)了刻畫第一光錐對偶曲面和第一雙曲對偶曲面的奇點(diǎn)的幾何不變量.最后,我們給出了一個(gè)具體的例子.第五章主要利用三維歐氏空間中的球面Legendrian對偶刻畫了正則曲線的球面指標(biāo)線之間的對偶關(guān)系.依據(jù)相對平行標(biāo)架場和奇點(diǎn)理論,解決了Bishop球面指標(biāo)線、Bishop球面Darboux像、Bishop對偶曲面及Bishop直紋曲面的奇點(diǎn)分類問題.我們還得到了Bishop斜螺線的一些性質(zhì).最后,我們給出了兩個(gè)具體例子.
[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.
【學(xué)位授予單位】：東北師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：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.
【學(xué)位授予單位】：東北師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O186.1
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