nLab
contact manifold

see also contact geometry

Idea

A contact manifold is a smooth manifold $P$ of odd dimension $2n+1$ which is equipped with a differential 1-form $A$ that is non-degenrate in the sense that the wedge product $A\wedge (d_{\mathrm{dR}}A{)}^{{\wedge }^n}$ does not vanish.

The special case of closed regular contact manifolds $(P,A)$ are essentially equivalent to the total spaces of circle bundles $P\to X$ over an $2n$-dimensional manifold equipped with a connection such that $A$ is the corresponding Ehresmann connection 1-form on the total space (BoothbyWang (1958)).

If in this case the curvature 2-form $\omega$ on $X$ makes the base space $X$ into a symplectic manifold, then $(P,A)$ is a corresponding prequantum circle bundle that provides a geometric prequantization of $(X,\omega )$.

A diffeomorphism $f:P\to P$ of a contact manifold $(P,A)$ is called a contactomorphism (in analogy with symplectomorphism) if it preserves the 1-form $A$ up to multiplication by a function. If $(P,A)$ 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 $(X,\omega )$.

Properties

Theorem

If $X$ is a closed smooth manifold, $P\to X$ a smooth circle bundle ($U(1)$-principal bundle) and $\omega \in {\Omega }^2(X)$ a differential 2-form representing its Chern class in de Rham cohomology, then there is a corresponding Ehresmann connection 1-form $A\in {\Omega }^1(P)$ with curvature $\omega$ and such that

1. $A$ is a regular contact form on $P$;

2. the Reeb vector field? of $A$ generates the given $U(1)$-action on $P$.

Moreover, every regular contact form $A$ on a closed manifold $P$ arises this way, up to rescaling by a constant.

This is due to (Boothby-Wang 58). The proof is recalled (and completed) in (Geiges 08, theorem 7.2.4, 7.2.5).

History

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 $M^{2n+1}$ can be reduced to the unitary group $U(n)$ and therefore all of its odd characteristic classes vanish. Since all the characteristic classes of a closed, orientable 3-manifold vanish, Chern posed the question of whether such a manifold always admits a contact structure and whehter there are non-isomorphic contact structures on one manifold in 1966.

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 ${ℝ}^3$. 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 ${ℝ}^3$ and $S^3$, the standard contact structure is the only tight contact structure.

Related concepts

• Legendrean submanifold

• symplectic manifold

• symplectic field theory

References

General

Monographs and introductions include

• Hansjörg Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics 109, Cambridge University Press, Cambridge, (2008) (pdf)

• 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)

Characterization

The observation that regular contact manifolds are prequantum circle bundles is due to

• W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958) 721–734. (JSTOR)

A modern review of this is in (Geiges, section 7.2).

An analogous result for a weaker form of regularity is discussed in

• C. Thomas, Almost regular contact manifolds, J. Diff. Geom. 11 (1976) (Euclid)

A characterization of $S^1$-invariant contact structures on circle bundles is in

• Fan Ding, Hansjörg Geiges, Contact structures on principal circle bundle, Bull. London Math. Soc, (arXiv:1107.4948)

For the special case of 2-dimensional base manifolds ($n=1$) this was obtained before in

• R. Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble) 27 (1977) no. 3, 1–15.

Further

See also

• Hansjörg Geiges, Contact structures on 1-connected 5-manifolds, Mathematika 38 (1991), 303-311; Contact structures on $(n-1)$-connected $(2n+1)$-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.