Spacelike Surfaces in Anti de Sitter Four-Space from a Contact Viewpoint

We define the notions of $(S_\mathrm t^1\times S_\mathrm s^2)$-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.