Spacelike surfaces in anti de Sitter four-space from a contact viewpoint. (English) Zbl 1201.53063

Izumiya, Shyuichi; Pei, Donghe; Del Carmen Romero Fuster, María

Proc. Steklov Inst. Math. 267, 156-173 (2009).

Summary: We define the notions of \((S_t ^{1} \times S_s ^{2})\)-nullcone Legendrian Gauss maps and \(S_{+}^{2}\)-nullcone Lagrangian Gauss maps on space-like 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 the 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.

MSC:

53C40	Global submanifolds
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C20	Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory

Keywords:

nullcone Legendrian Gauss maps; anti de Sitter 4-space; singularities; Lagrangian Gauss maps; Gauss-Bonnet type theorem; horospherical Gauss maps

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