1-lightlike Surfaces In Semi-euclidean 4-space With Index Two

In this thesis,by establishing some differential geometry theory on the 1-lightlike surfaces,we show several geometric properties of the 1-lightlike surfaces which are completely different from non-lightlike surfaces.Based on these theories,we consider the singularities of the 1-lightlike surfaces in semi-Euclidean 4-space with index two as an application of the theory of Legendrian singularities.We characterize the singularities of the 1-lightlike focal hypersurfaces and describe the contacts between the 1-lightlike surface and the anti de Sitter 3-sphere at singular points by employing Montaldi’s theory.In addition,we also discuss the detailed dif-ferential geometric properties of the 1-lightlike focal hypersurfaces in semi-Euclidean 4-space with index 2.The remainder of this thesis is organized as follows:We begin in Section 2 with the differential geometry of semi-Euclidean space with index two.In Section 3,we consider general 1-lightlike surfaces in semi-Euclidean space with index two and study their basic properties.We define the 1-lightlike distance-squared functions on a 1-lightlike surface and show that the discriminant set is a 1-lightlike focal hypersurface.In Section 4,we show further that the 1-lightlike distance-squared function of a 1-lightlike surface is a Morse family.In Section 5,we study the contact of a 1-lightlike surface with an anti de Sitter 3-sphere as an application of the theory of Legendrian singularities and discuss the geometric 1-lightlike surfaces in Section 6.Finally,an example will be proposed to explain our findings in Section 7.