The geometrical structure of 2D bond-orientational order

We study the formulation of bond-orientational order in an arbitrary two-di mensional geometry. We fmd that bond-orientational order is properly formul ated within the framework of differential geometry with torsion. The torsio n reflects the intrinsic frustration for two-dimensional crystals with arbi trary geometry. Within a Debye-Huckel approximation, torsion may be identif ied as the density of dislocations. Changes in the geometry of the system c ause a reorganization of the torsion density that preserves bond-orientatio nal order. As a byproduct, we are able to derive several identities involvi ng the topology, defect density and geometric invariants such as Gaussian c urvature. The formalism is used to derive the general free energy for a 2D sample of arbitrary geometry, in both the crystalline and hexatic phases. A pplications to conical and spherical geometries are briefly addressed.