Curves And Surfaces In Non-Euclidean Space

According to the sectional curvature,the Lorentzian space form is divided into three categories: de Sitter space with positive sectional curvature,Minkowski space with zero sectional curvature and Anti de Sitter space with negative sectional curvature.In chapter one,the basis notions in three dimensional Minkowski space will be firstly introduced,such as ruled surfaces generated by lightlike curves,the dual ruled surfaces etc.Then from the view point of differential geometry,we study the properties of the ruled surfaces generated by the lightlike rulings and the dual ruled surfaces.In particular,the characterizations of the dual ruled surfaces of B-scroll and null-scroll will be given respectively.In chapter two,the basis concepts of non-degenerate curves and non-degenerate surfaces in three dimensional de Sitter space will be firstly introduced.Moreover,the properties of differential geometry of principal normal geodesic surfaces which generated by the principal normal geodesics of the non-degenerate curves,such as the first spacelike,the timelike and the second spacelike,will be studied.At last,we get the relationship between Bertrand curves and the principal normal geodesic surfaces.