Spacelike surfaces in anti de Sitter four-space from a contact viewpoint

We define the notions of (S (t) (1) x S (s) (2) )-nullcone Legendrian Gauss maps and S (+) (2) -nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using S (+) (2) -nullcone Lagrangian Gauss maps, we define the notion of S (+) (2) -nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence, we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space, hyperbolic space, Lorentz-Minkowski space and de Sitter space.