  Smectic orders on curved substrates can be described by differential forms of
rank one (1-forms), whose geometric meaning is the differential of the local
phase field of density modulation. The exterior derivative of 1-form is the
local dislocation density. Elastic deformations are described by superposition
of exact differential forms. Applying this formalism to study smectic order on
torus as well as on sphere, we find that both systems exhibit many
topologically distinct low energy states, that can be characterized by two
integer topological charges. The total number of low energy states scales as
the square root of the substrate area. For smectic on a sphere, we also explore
the motion of disclinations as possible low energy excitations, as well as its
topological implications.
