Abstract
Anti de Sitter space is a maximally symmetric, vacuum solution of Einstein’s field equation with an attractive cosmological constant, and is the hyperquadric of semi-Euclidean space with index 2. So it is meaningful to study the submanifold in semi-Euclidean 4-space with index 2. However, the research on the submanifold in semi-Euclidean 4-space with index 2 has not been found from theory of singularity until now. In this paper, as a generalization of the study on lightlike hypersurface in Minkowski space and a preparation for the further study on anti de Sitter space, the singularities of lightlike hypersurface and Lorentzian surface in semi- Euclidean 4-space with index 2 will be studied. To do this, we reveal the relationships between the singularity of distance-squared function and that of lightlike hypersurface. In addition some geometric properties of lightlike hypersurface and Lorentzian surface are studied from geometrical point of view.
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