Special Curves And Special Ruled Surfaces In Anti De Sitter 3-space

Anti de Sitter space or Ads space for short is Lorentzian spatial model with negative sectional curvature -1. In Relativity Theory, Ads-space is one of the vacuum solutions in Einstein equations. Conducting the study of the curves and surfaces in Anti de Sitter Space not only has specific physical background, but also has profound theoretical meanings in mathematics. This paper will investigate special curves and special ruled surfaces in three dimensional Anti de Sitter space and some properties will be given.In the first chapter, the basic concepts and fundamental theorems of curves in three dimensional Anti de Sitter space will be introduced.In the second chapter, we give the definitions and some properties of principal normal geodesic surfaces, which is generated by principal normal geodesics of the first kind spacelike regular curves in three dimensional Anti de Sitter space. Moreover we define the first kind spacelike Bertrand curve in three dimensional Anti de Sitter space, some properties of the first kind spacelike Bertrand curve will be shown, and the relationship between the first kind spacelike Bertrand curves and the geodesic surfaces of the first kind spacelike regular curves in three dimensional Anti de Sitter space will be given.In the third chapter, we give the definitions and some properties of the principal normal geodesic surfaces, which is generated by principal normal geodesics of regular timelike curves in three dimensional Anti de Sitter space. And then, we define timelike Bertrand curves in three dimensional Anti de Sitter space, some properties of the timelike Bertrand curves will be studied. Moreover the relationship between the timelike Bertrand curves and the geodesic surfaces of the timelike regular curves in three dimensional Anti de Sitter space will also be given.In the fourth chapter, we introduce the definition of null Cartan curves and the rel-evant definition of null Cartan surfaces in 3-dimensional Anti de Sitter space. We define the height function of the null Cartan curve and introduce two new geometric invariants.By using the unfolding theory of functions in singularity theory and those invariants, the classification of singularities of the null Cartan surface is given and the geometric properties of the singularities are described.