nLab contact manifold

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-degenerate in the sense that the wedge product $A \wedge (d_{dR} A)^{\wedge^n}$ 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

Darboux theorem

There is a Darboux theorem for contact structures, stating how they are locally equivalent to a standard contact structure (e.g. Arnold 78, page 362)

Relation to $U(1)$-principal connections

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 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 $\mathbb{R}^3$ (Bennequin 83). Bennequin recognized that the induced singular foliation on a surface embedded in a contact 3-manifold plays a crucial role for the classification 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 classification for overtwisted contact structures on 3-manifolds (Eliashberg 89). Eliashberg also proved that on $\mathbb{R}^3$ and $S^3$, the standard contact structure is the only tight contact structure.

Related concepts

• Reeb vector field

• Legendrean submanifold

• symplectic manifold

• symplectic field theory

• generalized contact geometry

• Sasakian manifold

References

General

Monographs and introductions:

• Hansjörg Geiges, Contact geometry, in F. J. E. Dillen and L. C. A. Verstraelen, (eds.), Handbook of Differential Geometry vol. 2 North-Holland, Amsterdam (2006) 315-382 [arXiv:math/0307242, ISBN:978-0-444-52052-4]

• Hansjörg Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics 109, Cambridge University Press (2008) [doi:10.1017/CBO9780511611438]

• 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]

Discussion in terms of (a modification of) G-structures:

• Alfonso Giuseppe Tortorella, Luca Vitagliano & Ori Yudilevich, Homogeneous G-structures, Annali di Matematica (2020) [doi:10.1007/s10231-020-00972-9]

A higher differential geometry-generalization of contact geometry in line with multisymplectic geometry/n-plectic geometry is discussed in

• Luca Vitagliano, L-infinity Algebras From Multicontact Geometry (arXiv.1311.2751)

• Vladimir Arnol'd, Mathematical methods of classical mechanics, Graduate texts in Mathematics 60 (1978)

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:10.2307/1970165)

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:jdg/1214433722)

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.

See also

• John Bland, Tom Duchamp, The Group of Contact Diffeomorphisms for Compact Contact Manifolds (arXiv:1007.2036)

Further

• Daniel Bennequin, Entrelacements et equations de Pfaff, Astérique, 107-108 (1983), 87-161. (pdf)

• Yakov Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), 623-637. (pdf)

• Yakov Eliashberg, Contact 3-manifolds twenty years since J Martinet’s work, Ann. Inst. Fourier, 42 (1992), 165-192. (pdf)

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

• Hansjörg Geiges, Contact structures on $(n-1)$-connected $(2n+1)$-manifolds, Pacific J. Math. 161 (1993), 129-137;

• Hansjörg Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), 455-464;

• Hansjörg Geiges, 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.