  The uniform director field obtained for the nematic ground state of the
hard-rod model of liquid crystals in two dimensions reflects the high symmetry
of the constituents of the liquid; It is a manifestation of the constituents'
local tendency to avoid splaying and bending with respect to one another. In
contrast, bent-core (or banana shaped) liquid-crystal-forming-molecules locally
favor a state of zero splay and constant bend. However, such a structure cannot
be realized in the plane and the resulting liquid-crystalline phase is
frustrated and must exhibit some compromise of these two mutually contradicting
local intrinsic tendencies. The generation of geometric frustration from the
intrinsic geometry of the constituents of a material is not only natural and
ubiquitous but also leads to a striking variety of morphologies of ground
states and exotic response properties.
  In this work we establish the necessary and sufficient conditions for two
scalar functions, $s$ and $b$ to describe the splay and bend of a director
field in the plane. We generalize these compatibility conditions for geometries
with non-vanishing constant Gaussian curvature, and provide a reconstruction
formula for the director field depending only on the splay and bend fields and
their derivatives. Last, we discuss optimal compromises for simple incompatible
cases where the locally preferred values of the splay and bend cannot be
globally achieved.
