1-Lightlike surfaces in semi-Euclidean 4-space with index two

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 differential geometric properties of the 1-lightlike focal hypersurfaces in semi-Euclidean 4-space with index 2. Finally, an example will be proposed to explain our findings.