In this paper,we mainly research the local differential geometry nature of a Lorentzian Hypersurface in Anti de Sitter n-Space, and define the notions of St1×Ssn-1-valude light-cone Gauss maps and St1×Ssn-1-valude lightcone height function, also we structure the lightcone pedal surface, prove the relationship between the singularity of the lightcone pedal surface and the singularity of the lightcone Gauss map which offer the theoretical foundation for singularities of the Lorentzian hypersurfaee using Legendrian theory in future.
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