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 {ÃÂ¥it
Anti de Sitter nullcone Gauss image} and {ÃÂ¥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 {ÃÂ¥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 ÃÂ¥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 ÃÂ¥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.