  Conical surfaces, with a delta function of Gaussian curvature at the apex,
are perhaps the simplest example of geometric frustration. We study
two-dimensional liquid crystals with $p$-fold rotational symmetry ($p$-atics)
on the surfaces of cones. For free boundary conditions at the base, we find
both the ground state(s) and a discrete ladder of metastable states as a
function of both the cone angle and the liquid crystal symmetry $p$. We find
that these states are characterized by a set of fractional defect charges at
the apex and that the ground states are in general frustrated due to effects of
parallel transport along the azimuthal direction of the cone. We check our
predictions for the ground state energies numerically for a set of commensurate
cone angles (corresponding to a set of commensurate Gaussian curvatures
concentrated at the cone apex), whose surfaces can be polygonized as a perfect
triangular or square mesh, and find excellent agreement with our theoretical
predictions.
