  We use the method of $\Gamma$-convergence to study the behavior of the
Landau-de Gennes model for a nematic liquid crystalline film attached to a
general fixed surface in the limit of vanishing thickness. This paper
generalizes the approach that we used previously to study a similar problem for
a planar surface. Since the anchoring energy dominates when the thickness of
the film is small, it is essential to understand its influence on the structure
of the minimizers of the limiting energy. In particular, the anchoring energy
dictates the class of admissible competitors and the structure of the limiting
problem. We assume general weak anchoring conditions on the top and the bottom
surfaces of the film and strong Dirichlet boundary conditions on the lateral
boundary of the film when the surface is not closed. We establish a general
convergence result to an energy defined on the surface that involves a somewhat
surprising remnant of the normal component of the tensor gradient. Then we
exhibit one effect of curvature through an analysis of the behavior of
minimizers to the limiting problem when the substrate is a frustrum.
