On Local Differential Geometry Of Submanifolds In Anti De Sitter 3-space

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.