  We investigate the geometric properties of marginally trapped surfaces
(surfaces which have null mean curvature vector) in the spaces of oriented
geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their
canonical neutral Kaehler structures.
  We prove that every rank one surface in these four manifolds is marginally
trapped. In the Euclidean case we show that Lagrangian rotationally symmetric
sections are marginally trapped and construct an explicit family of marginally
trapped Lagrangian tori.
  In the hyperbolic case we explore the relationship between marginally trapped
and Weingarten surfaces, and construct examples of marginally trapped surfaces
with various properties.
