Topology and Geometry of Smectic Order on Compact Curved Substrates

We show that smectic order on arbitrary curved substrate can be described by differential form of rank one (1-form), whose geometric meaning is the differential of the local phase field of the density modulation. The exterior derivative of 1-form is shown to be the local dislocation density. Elastic deformations are described by superposition of exact differential forms. More importantly, we use the formalism of differential forms to systematically classify and characterize all low energy smectic states on torus as well as on sphere. Two dimensional smectic systems confined on either manifold exhibit many topologically distinct low energy states. Different states are not accessible from each other by local fluctuations. The total number of low energy states scales as the square root of the substrate area. We also address the energetics of 2D smectics on curved substrate and calculate the mean field phase diagram of smectics on a thin torus. Finally, we also discuss the motion of disclinations for spherical smectics, and illustrate the interesting connection between spherical smectics and the theory of elliptic functions.