  The topology and the geometry of a surface play a fundamental role in
determining the equilibrium configurations of thin films of liquid crystals. We
propose here a theoretical analysis of a recently introduced surface Frank
energy, in the case of two-dimensional nematic liquid crystals coating a
toroidal particle. Our aim is to show how a different modeling of the effect of
extrinsic curvature acts as a selection principle among equilibria of the
classical energy, and how new configurations emerge. In particular, our
analysis predicts the existence of new stable equilibria with complex windings.
