Dept. Math, Hokkaido Univ. EPrints Server

Doctoral thesis "On differential geometry of surfaces in anti de Sitter 3-space''

Technical Report # 144
Chen, Liang Doctoral thesis "On differential geometry of surfaces in anti de Sitter 3-space''. (11 February 2010);

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Abstract

We investigate the differential geometry of surfaces in Anti de Sitter 3-space as an application of the theory of singularities. We first show six ¥textit{Legendrian dualities} between pseudo-spheres in semi-Euclidean 4-space with index 2 which are basic tools for the study of extrinsic differential geometry of submanifolds in these pseudo-spheres. Secondly, we apply the Legendrian dualities to investigate the geometric properties of non-degenerate surfaces in Anti de Sittter 3-space. We study the spacelike surfaces in Anti de Sitter 3-space as the first step of this research. We define the timelike Anti de Sitter Gauss images and timelike Anti de Sitter height functions on spacelike surfaces and investigate the geometric meanings of singularities of these mappings. We consider the contact of spacelike surfaces with models (so-called AdS-great-hyperboloids) as an application of Legendrian singularity theory. Thirdly, we also use the Legendrian dualities to study the geometric properties of timelike surfaces in Anti de Sitter 3-space. We define two mappings associated to a timelike surface which are called {Â¥it Anti de Sitter nullcone Gauss image} and {Â¥it Anti de Sitter torus Gauss map.} We can prove that the nullcone Gauss image is a Legendrian map and the classification of its generic singularities is given. We investigate the relation between the Anti de Sitter null Gauss image and the Anti de Sitter torus Gauss mapping. We prove that the Anti de Sitter torus Gauss mapping is a Lagrangian mapping, and that the Legendrian lift of the Anti de Sitter nullcone Gauss image is a covering of the Lagrangian lift of the Anti de Sitter torus Gauss mapping. We also define a family of functions named {Â¥it Anti de Sitter null height function} on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map. At last, we study the geometric properties of degenerate surfaces, which are called Â¥textit{AdS null surfaces}, in Anti de Sitter 3-space from a contact viewpoint. These surfaces are associated to spacelike curves in Anti de Sitter 3-space. The geometry of these spacelike curves determines the behavior of the corresponding AdS null surfaces. We define a map which is called the Â¥textit{torus Gauss image}. It appears to be associated to the contacts of the spaclike curve with some model in Anti de Sitter 3-space. We also define two families of functions and use them to investigate the singularities of AdS null surfaces and torus Gauss images as applications of singularity theory of functions.

Item Type:Technical Report
Uncontrolled Keywords:Anti de Sitter 3-space; Legendrian dualities; Legendrian singularities; spacelike surface; timelike surface; AdS-null surface; spacelike curve; versal unfolding
Subjects:53-xx DIFFERENTIAL GEOMETRY
ID Code:2033