3维Anti de Sitter空间中子流形的局部微分几何
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摘要
众所周知，Anti de Sitter空间(或，AdS-空间)是具有负的常截面曲率的Lorentzian空间型，是理论物理学中一个很重要的研究对象，在相对论中，AdS-空间是Einstein方程的真空解之一，且具有正的能量表示．在物理学中有一个猜想：AdS-空间上的经典重力理论与这个空间理想边界上的共形场理论是等价的．E．Witten称这个猜想为AdS／CFT对应(Anti de Sitter space／conformal field the-ory correspondence)或AdS-全息术原理(AdS-holography principle)．这个对应为研究量子重力和量子色动力提供了很好的研究思路，是理论物理中一个很重要的概念性突破之一．如果用数学的语言来解释AdS／CFT对应则为：AdS-空间中子流形的微分几何性质与AdS-空间中理想边界上的规范场论的几何性质是相对应的．因此对AdS-空间中子流形的微分几何性质进行研究是很有意义的．然而，目前对AdS-空间中浸入子流形的几何性质的研究还不多，尤其是从奇点理论的角度．近年来，应用奇点理论对各类空间型中子流形几何性质的研究得到了飞速发展，特别是S．Izumiya教授和裴东河教授等做出了很多理想的结果．实践证明，无论从局部还是从整体的角度来看，奇点理论都是研究不同空间中浸入子流形几何性质的一个强有力工具．奇点与几何之间的自然联系反应了子流形与某些模型(即合适变换群作用下的不变量)之间的切触关系．本文则以奇点理论为工具对3维AdS-空间中的子流形与某些模型曲面之间的切触关系进行了深入细致的研究．
第一章是引言部分．我们介绍了奇点理论的发展史以及奇点理论在应用方面的最新成果，同时还介绍了本文的基本框架．
第二章介绍了指标为2的半欧氏空间和切触几何的相关知识．特别地，我们证明了指标为2的半欧氏空间中伪球间的Legendrian对偶定理(定理2．2．1和定理2．2．2)，这些定理是Izumiya结果的推广．利用这些定理我们可以对3维Anti de Sitter空间中非退化曲面上的高斯映射的性态进行刻画．
第三章构建了3维AdS-空间中类空曲面的基本框架．在类空曲面上定义类时Anti de Sitter高斯像(简记为，TAdS-高斯像)和类时Anti de Sitter高度函数(简记为，AdS-高度函数)，并且研究这些映射奇点的几何意义．作为Legendrian奇点理论的应用，考虑类空曲面与模型曲面(即，平坦AdS-双曲面)的切触问题．
第四章从切触的观点考虑3维AdS-空间中的类时曲面．定义两个与类时曲面相关的映射，分别称为Anti de Sitter nullcone高斯像(简记为，AdS-nullcone高斯像)和Anti de Sitter环面高斯映射(简记为，AdS-环面高斯映射)．在类时曲面上定义一个函数族，称之为Anti de Sitter null高度函数．把这个函数族作为基本工具来研究AdS-nullcone高斯像和AdS-环面高斯映射的奇点的几何意义．我们也考虑类时曲面与模型曲面(即，AdS-极限球)的切触关系．尽管研究方法是类似的，但是在类时曲面情形可以得到很多新的几何信息．
第五章研究3维AdS-空间中退化曲面Anti de Sitter null曲面(简记为，AdS-null曲面)的几何性质．这个曲面与3维AdS-空间中的类空曲线相关联．我们在类空曲线上定义一个映射并称之为环面高斯像；又定义两个函数族，分别称为类空曲线上的环面高度函数和AdS-距离平方函数．作为函数奇点理论的应用，利用这两个函数族来研究AdS-null曲面和环面高斯像的奇点．
第六章主要研究3维AdS-空间中的类时曲线．但是，正如我们所期待的，此时的情形与类空曲线的情形相比较是很特别的．我们可以构造一个与3维AdS-空间中的类时曲线相关联的3维nullcone空间中的类空曲面．我们利用函数通用开折理论研究这个类空曲面的奇点的几何意义．
As been well known, the Lorentzian space form with the constant negative curvatureis called Anti de Sitter space(or, AdS-space). This space is a very importantsubject in physics, it is also one of the vacuum solutions of the Einstein equation in thetheory of relativity. There is a conjecture in physics that the classical gravitation theoryon AdS-space is equivalent to the conformal field theory on the ideal boundary ofAdS-space. It is called the AdS/CFT-correspondence or the holographic principle byE. Witten. In mathematics this conjecture is that the extrinsic geometric properties onsubmanifolds in AdS-space have corresponding Gauge theoretic geometric propertiesin its ideal boundary. Therefore, it is very important to investigate the geometric propertiesof submanifolds immersed in AdS-space. However, there are not many resultson submanifolds in AdS-space, in particular from the view point of singularity theory.Singularity theory tools have proven to be useful in the description of geometric propertiesof submanifolds immersed in different ambient spaces, from both the local andglobal viewpoint. Recently, the geometric properties of submanifolds immersed in differentspace forms had been well developed. Especially, professors S. Izumiya and D.Pei et al. have got many excellent results. The natural connection between Geometryand Singularities is the contacts of the submanifolds with the models (invariant underthe action of a suitable transformation group) of the ambient space. In this paper, weinvestigate the local differential geometry of submanifolds in Anti de Sitter 3-space asapplications of singularity theory.
The introduction is located in Chapter one. We introduce the history of singularitytheory and the new results of the applications of it. Also we introduce the basic frameof this paper.
In Chapter two, we show the basic notions on semi-Euclidean space with index2 and contact geometry. Especially we have proved the Legendrian duality theorem(Theorem 2.2.1 and Theorem 2.2.2) between pseudo-spheres in semi-Euclidean spacewith index 2, which are generalizations of the previous results of Izumiya.
In Chapter three, we construct a basic framework for the study of spacelike surfacesin Anti de Sitter 3-space. We define a timelike Anti de Sitter Gauss image (briefly,TAdS-Gauss image) and a timelike Anti de Sitter height function (briefly, AdS-heightfunction) on the spacelike surface and investigate the geometric meanings of singular- ities of these mappings. We consider the contact of the spacelike surfaces with models(so-called AdS-flat-hyperboloids) as an application of Legendrian singularity theory.
In Chapter four, we investigate timelike surfaces in Anti de Sitter 3-space fromthe viewpoint of contact. We define two mappings associated to a timelike surfacewhich are called an Anti de Sitter nullcone Gauss image (briefly, AdS-nullcone Gaussimage) and an Anti de Sitter torus Gauss map (briefly, AdS-torus Gauss map). Wealso define a family of functions named an Anti de Sitter null height function on thetimelike surface. We use this family of functions as a basic tool to investigate thegeometric meanings of singularities of the AdS-nullcone Gauss image and the AdS-torusGauss map. Also we consider the contact of the timelike surfaces with models(so-called AdS-horospheres).
In Chapter five, we study the geometric properties of degenerate surfaces, whichare called the AdS-null surfaces in Anti de Sitter 3-space. These surfaces are associatedto spacelike curves in Anti de Sitter 3-space. We define a map which is called a torusGauss image. We also define two families of functions, named a torus height functionand an AdS-distance-squared function respectively, and use them to investigate thesingularities of the AdS-null surfaces and the torus Gauss images as applications ofsingularity theory of functions.
In Chapter six, we consider timelike curves in Anti de Sitter 3-space by exactlythe same arguments as those of Chapter five. However, as it was to be expected, thesituation presents certain peculiarities when compared with the spacelike curve case.In this case, we can construct a spacelike surface in nullcone 3-space associated to atimelike curve in Anti de Sitter 3-space. We study the geometric meanings of singularitiesof this spacelike surface as an application of versal unfolding theory of functions.
第一章是引言部分．我们介绍了奇点理论的发展史以及奇点理论在应用方面的最新成果，同时还介绍了本文的基本框架．
第二章介绍了指标为2的半欧氏空间和切触几何的相关知识．特别地，我们证明了指标为2的半欧氏空间中伪球间的Legendrian对偶定理(定理2．2．1和定理2．2．2)，这些定理是Izumiya结果的推广．利用这些定理我们可以对3维Anti de Sitter空间中非退化曲面上的高斯映射的性态进行刻画．
第三章构建了3维AdS-空间中类空曲面的基本框架．在类空曲面上定义类时Anti de Sitter高斯像(简记为，TAdS-高斯像)和类时Anti de Sitter高度函数(简记为，AdS-高度函数)，并且研究这些映射奇点的几何意义．作为Legendrian奇点理论的应用，考虑类空曲面与模型曲面(即，平坦AdS-双曲面)的切触问题．
第四章从切触的观点考虑3维AdS-空间中的类时曲面．定义两个与类时曲面相关的映射，分别称为Anti de Sitter nullcone高斯像(简记为，AdS-nullcone高斯像)和Anti de Sitter环面高斯映射(简记为，AdS-环面高斯映射)．在类时曲面上定义一个函数族，称之为Anti de Sitter null高度函数．把这个函数族作为基本工具来研究AdS-nullcone高斯像和AdS-环面高斯映射的奇点的几何意义．我们也考虑类时曲面与模型曲面(即，AdS-极限球)的切触关系．尽管研究方法是类似的，但是在类时曲面情形可以得到很多新的几何信息．
第五章研究3维AdS-空间中退化曲面Anti de Sitter null曲面(简记为，AdS-null曲面)的几何性质．这个曲面与3维AdS-空间中的类空曲线相关联．我们在类空曲线上定义一个映射并称之为环面高斯像；又定义两个函数族，分别称为类空曲线上的环面高度函数和AdS-距离平方函数．作为函数奇点理论的应用，利用这两个函数族来研究AdS-null曲面和环面高斯像的奇点．
第六章主要研究3维AdS-空间中的类时曲线．但是，正如我们所期待的，此时的情形与类空曲线的情形相比较是很特别的．我们可以构造一个与3维AdS-空间中的类时曲线相关联的3维nullcone空间中的类空曲面．我们利用函数通用开折理论研究这个类空曲面的奇点的几何意义．
As been well known, the Lorentzian space form with the constant negative curvatureis called Anti de Sitter space(or, AdS-space). This space is a very importantsubject in physics, it is also one of the vacuum solutions of the Einstein equation in thetheory of relativity. There is a conjecture in physics that the classical gravitation theoryon AdS-space is equivalent to the conformal field theory on the ideal boundary ofAdS-space. It is called the AdS/CFT-correspondence or the holographic principle byE. Witten. In mathematics this conjecture is that the extrinsic geometric properties onsubmanifolds in AdS-space have corresponding Gauge theoretic geometric propertiesin its ideal boundary. Therefore, it is very important to investigate the geometric propertiesof submanifolds immersed in AdS-space. However, there are not many resultson submanifolds in AdS-space, in particular from the view point of singularity theory.Singularity theory tools have proven to be useful in the description of geometric propertiesof submanifolds immersed in different ambient spaces, from both the local andglobal viewpoint. Recently, the geometric properties of submanifolds immersed in differentspace forms had been well developed. Especially, professors S. Izumiya and D.Pei et al. have got many excellent results. The natural connection between Geometryand Singularities is the contacts of the submanifolds with the models (invariant underthe action of a suitable transformation group) of the ambient space. In this paper, weinvestigate the local differential geometry of submanifolds in Anti de Sitter 3-space asapplications of singularity theory.
The introduction is located in Chapter one. We introduce the history of singularitytheory and the new results of the applications of it. Also we introduce the basic frameof this paper.
In Chapter two, we show the basic notions on semi-Euclidean space with index2 and contact geometry. Especially we have proved the Legendrian duality theorem(Theorem 2.2.1 and Theorem 2.2.2) between pseudo-spheres in semi-Euclidean spacewith index 2, which are generalizations of the previous results of Izumiya.
In Chapter three, we construct a basic framework for the study of spacelike surfacesin Anti de Sitter 3-space. We define a timelike Anti de Sitter Gauss image (briefly,TAdS-Gauss image) and a timelike Anti de Sitter height function (briefly, AdS-heightfunction) on the spacelike surface and investigate the geometric meanings of singular- ities of these mappings. We consider the contact of the spacelike surfaces with models(so-called AdS-flat-hyperboloids) as an application of Legendrian singularity theory.
In Chapter four, we investigate timelike surfaces in Anti de Sitter 3-space fromthe viewpoint of contact. We define two mappings associated to a timelike surfacewhich are called an Anti de Sitter nullcone Gauss image (briefly, AdS-nullcone Gaussimage) and an Anti de Sitter torus Gauss map (briefly, AdS-torus Gauss map). Wealso define a family of functions named an Anti de Sitter null height function on thetimelike surface. We use this family of functions as a basic tool to investigate thegeometric meanings of singularities of the AdS-nullcone Gauss image and the AdS-torusGauss map. Also we consider the contact of the timelike surfaces with models(so-called AdS-horospheres).
In Chapter five, we study the geometric properties of degenerate surfaces, whichare called the AdS-null surfaces in Anti de Sitter 3-space. These surfaces are associatedto spacelike curves in Anti de Sitter 3-space. We define a map which is called a torusGauss image. We also define two families of functions, named a torus height functionand an AdS-distance-squared function respectively, and use them to investigate thesingularities of the AdS-null surfaces and the torus Gauss images as applications ofsingularity theory of functions.
In Chapter six, we consider timelike curves in Anti de Sitter 3-space by exactlythe same arguments as those of Chapter five. However, as it was to be expected, thesituation presents certain peculiarities when compared with the spacelike curve case.In this case, we can construct a spacelike surface in nullcone 3-space associated to atimelike curve in Anti de Sitter 3-space. We study the geometric meanings of singularitiesof this spacelike surface as an application of versal unfolding theory of functions.
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