  We discuss three mathematical structures which arise in topologically massive
abelian gauge theory. First, the euclidean topologically massive abelian gauge
theory defines a contact structure on a manifold. We briefly discuss three
solutions and the related contact structures on the flat 3-torus, the AdS
space, the 3-sphere which respectively correspond to Bianchi type I, VIII, IX
spaces. We also present solutions on Bianchi type II, VI and VII spaces.
Secondly, we discuss a family of complex (anti-)self-dual solutions of the
euclidean theory in cartesian coordinates on R3 which are given by
(anti-)holomorpic functions. The orthogonality relation of contact structures
which are determined by the real parts of these complex solutions separates
them into two classes: the self-dual and the anti-self-dual solutions. Thirdly,
we apply the curl transformation to this theory. An arbitrary solution is given
by a vector tangent to a sphere whose radius is determined by the topological
mass in transform space. Meanwhile a gauge transformation corresponds to a
vector normal to this sphere. We discuss the quantization of topological mass
on an example.
