  We investigate a counterintuitive geometric interaction between defects and
curvature in thin layers of superfluids, superconductors and liquid crystals
deposited on curved surfaces. Each defect feels a geometric potential whose
functional form is determined only by the shape of the surface, but whose sign
and strength depend on the transformation properties of the order parameter.
For superfluids and superconductors, the strength of this interaction is
proportional to the square of the charge and causes all defects to be repelled
(attracted) by regions of positive (negative) Gaussian curvature. For liquid
crystals in the one elastic constant approximation, charges between 0 and
$4\pi$ are attracted by regions of positive curvature while all other charges
are repelled.
