see also contact geometry
The special case of closed regular contact manifolds are essentially equivalent to the total spaces of circle bundles over an -dimensional manifold equipped with a connection such that is the corresponding Ehresmann connection 1-form on the total space (BoothbyWang (1958)).
A diffeomorphism of a contact manifold is called a contactomorphism (in analogy with symplectomorphism) if it preserves the 1-form up to multiplication by a function. If is regular and hence a prequantum line bundle a contactomorphism that strictly preserves the connection 1-form is called a quantomorphism. The Lie algebra of the quantomorphism group is that of the Poisson algebra of the base symplectic manifold .
If is a closed smooth manifold, a smooth circle bundle (-principal bundle) and a differential 2-form representing its Chern class in de Rham cohomology, then there is a corresponding Ehresmann connection 1-form with curvature and such that
Moreover, every regular contact form on a closed manifold arises this way, up to rescaling by a constant.
The following is taken from (Lin).
Originating in Hamiltonian mechanics and geometric optics, contact geometry caught geometers’ attention much earlier. In 1953, Shiing-shen Chern showed that the structure group of a contact manifold can be reduced to the unitary group and therefore all of its odd characteristic classes vanish. Since all the characteristic classes of a closed, orientable 3-manifold vanish, Chern in 1966 posed the questions of whether such a manifold always admits a contact structure and whether there are non-isomorphic contact structures on one manifold.
One of the milestones in the study of contact geometry is Bennequin’s proof of the existence of exotic contact structures (i. e., contact structures not isomorphic to the standard one) on . Bennequin recognized that the induced singular foliation on a surface embedded in a contact 3-manifold plays a crucial role for the classication of contact structures. This role was further explored in the work of Eliashberg, who distinguished two classes of contact structures in dimension 3, overtwisted and tight, and gave a homotopy classication for overtwisted contact structures on 3-manifolds. Eliashberg also proved that on and , the standard contact structure is the only tight contact structure.
Monographs and introductions include
Xiao-Song Lin, An introduction to 3-dimensional contact geometry (pdf)
John Etnyre, Introductory lectures on contact geometry Proc. Sympos. Pure Math. 71 (2003), 81-107. (pdf)
The observation that regular contact manifolds are prequantum circle bundles is due to
A modern review of this is in (Geiges, section 7.2).
An analogous result for a weaker form of regularity is discussed in
A characterization of -invariant contact structures on circle bundles is in
For the special case of 2-dimensional base manifolds () this was obtained before in
Hansjörg Geiges, Contact structures on 1-connected 5-manifolds, Mathematika 38 (1991), 303-311; Contact structures on -connected -manifolds, Pacific J. Math. 161 (1993), 129-137; Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), 455-464; Normal contact structures on 3-manifolds, Tôhoku Math. J. 49 (1997), 415-422.
Hansjörg Geiges, J. Gonzalo, Moduli of contact circles, J. Reine Angew. Math. 551 (2002), 41-85; Contact geometry and complex surfaces, Invent. Math. 121 (1995), 147-209.