The standard contact structure on R³. Each point in R³ has a plane associated to it by the contact structure, in this case as the kernel of the one-form dz − y dx. These planes appear to twist along the y-axis. It is not integrable, as can be verified by drawing an infinitesimal square in the x-y plane, and follow the path along the one-forms. The path would not return to the same z-coordinate after one circuit.

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension.

Applications

Like symplectic geometry, contact geometry has broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, integrable systems and to control theory. Contact geometry also has applications to low-dimensional topology; for example, it has been used by Kronheimer and Mrowka to prove the property P conjecture, by Michael Hutchings to define an invariant of smooth three-manifolds, and by Lenhard Ng to define invariants of knots. It was also used by Yakov Eliashberg to derive a topological characterization of Stein manifolds of dimension at least six.

Contact forms and structures

A contact structure on an odd dimensional manifold is a smoothly varying family of codimension one subspaces of each tangent space of the manifold, satisfying a non-integrability condition. The family may be described as a section of a bundle as follows:

Given an n-dimensional smooth manifold M, and a point p ∈ M, a contact element of M with contact point p is an (n − 1)-dimensional linear subspace of the tangent space to M at p.^[1][2] A contact element can be given by the kernel of a linear function on the tangent space to M at p. However, if a subspace is given by the kernel of a linear function ω, then it will also be given by the zeros of λω where λ ≠ 0 is any nonzero real number. Thus, the kernels of { λω : λ ≠ 0 } all give the same contact element. It follows that the space of all contact elements of M can be identified with a quotient of the cotangent bundle T*M (with the zero section ${\displaystyle 0_{M}}$ removed),^[1] namely:

${\displaystyle {\text{PT}}^{*}M=({\text{T}}^{*}M-{0_{M}})/\!\sim \ {\text{ where, for }}\omega _{i}\in {\text{T}}_{p}^{*}M,\ \ \omega _{1}\sim \omega _{2}\ \iff \ \exists \ \lambda \neq 0\$ :\ \omega _{1}=\lambda \omega _{2}.}

A contact structure on an odd dimensional manifold M, of dimension 2k+1, is a smooth distribution of contact elements, denoted by ξ, which is generic at each point.^[1][2] The genericity condition is that ξ is non-integrable.

Assume that we have a smooth distribution of contact elements, ξ, given locally by a differential 1-form α; i.e. a smooth section of the cotangent bundle. The non-integrability condition can be given explicitly as:^[1]

${\displaystyle \alpha \wedge ({\text{d}}\alpha )^{k}\neq 0\ {\text{where}}\ ({\text{d}}\alpha )^{k}=\underbrace {{\text{d}}\alpha \wedge \ldots \wedge {\text{d}}\alpha } _{k-{\text{times}}}.}$

Notice that if ξ is given by the differential 1-form α, then the same distribution is given locally by β = ƒ⋅α, where ƒ is a non-zero smooth function. If ξ is co-orientable then α is defined globally.

Legendrian submanifolds and knots

The most interesting subspaces of a contact manifold are its Legendrian submanifolds. The non-integrability of the contact hyperplane field on a (2n + 1)-dimensional manifold means that no 2n-dimensional submanifold has it as its tangent bundle, even locally. However, it is in general possible to find n-dimensional (embedded or immersed) submanifolds whose tangent spaces lie inside the contact field: these are called Legendrian submanifolds.

Legendrian submanifolds are analogous to Lagrangian submanifolds of symplectic manifolds. There is a precise relation: the lift of a Legendrian submanifold in a symplectization of a contact manifold is a Lagrangian submanifold.

The simplest example of Legendrian submanifolds are Legendrian knots inside a contact three-manifold. Inequivalent Legendrian knots may be equivalent as smooth knots; that is, there are knots which are smoothly isotopic where the isotopy cannot be chosen to be a path of Legendrian knots.

Legendrian submanifolds are very rigid objects; typically there are infinitely many Legendrian isotopy classes of embeddings which are all smoothly isotopic. Symplectic field theory provides invariants of Legendrian submanifolds called relative contact homology that can sometimes distinguish distinct Legendrian submanifolds that are topologically identical (i.e. smoothly isotopic).

Reeb vector field

If α is a contact form for a given contact structure, the Reeb vector field R can be defined as the unique element of the (one-dimensional) kernel of dα such that α(R) = 1. If a contact manifold arises as a constant-energy hypersurface inside a symplectic manifold, then the Reeb vector field is the restriction to the submanifold of the Hamiltonian vector field associated to the energy function. (The restriction yields a vector field on the contact hypersurface because the Hamiltonian vector field preserves energy levels.)

The dynamics of the Reeb field can be used to study the structure of the contact manifold or even the underlying manifold using techniques of Floer homology such as symplectic field theory and, in three dimensions, embedded contact homology. Different contact forms whose kernels give the same contact structure will yield different Reeb vector fields, whose dynamics are in general very different. The various flavors of contact homology depend a priori on the choice of a contact form, and construct algebraic structures the closed trajectories of their Reeb vector fields; however, these algebraic structures turn out to be independent of the contact form, i.e. they are invariants of the underlying contact structure, so that in the end, the contact form may be seen as an auxiliary choice. In the case of embedded contact homology, one obtains an invariant of the underlying three-manifold, i.e. the embedded contact homology is independent of contact structure; this allows one to obtain results that hold for any Reeb vector field on the manifold.

The Reeb field is named after Georges Reeb.

Some historical remarks

The roots of contact geometry appear in work of Christiaan Huygens, Isaac Barrow, and Isaac Newton. The theory of contact transformations (i.e. transformations preserving a contact structure) was developed by Sophus Lie, with the dual aims of studying differential equations (e.g. the Legendre transformation or canonical transformation) and describing the 'change of space element', familiar from projective duality.

The first known use of the term "contact manifold" appears in a paper of 1958^[3][4][5]

See also

• Floer homology, some flavors of which give invariants of contact manifolds and their Legendrian submanifolds
• Quantized contact transformation
• Sub-Riemannian geometry

References

External links

• Contact manifold at the Manifold Atlas