Isotropy and marginally trapped surfaces in a spacetime

In this paper we shall study the notions of an isotropic and marginally trapped surface in a spacetime by using a differential geometric approach. We first consider spacelike isotropic surfaces in a Lorentzian manifold and, in particular, in a four-dimensional spacetime, where the isotropy function appears to be determined by the mean curvature vector field of the surface. Explicit examples of isotropic marginally outer trapped surfaces are given in the standard four-dimensional space forms: Minkowski, de Sitter and anti-de Sitter spaces. Then we prove rigidity theorems for complete spacelike 0-isotropic surfaces without flat points in these standard space forms. As a consequence, we also obtain characterizations of complete spacelike isotropic marginally trapped surfaces in these backgrounds.