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Wikipedia / Contact geometry

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.

Mathematical formulation

Contact structure

-dimensional smooth manifold, a contact element-dimensional linear subspace of the tangent space. A contact structure, is a smooth distribution, which is generic (in the sense of being maximally non-integrable) at each point. A contact manifold is a smooth manifold equipped with a contact structure.

can be identified with a quotient of the cotangent bundle removed), namely:.

-th contact bundle.

By Darboux's theorem, all contact structures of the same dimension are locally diffeomorphic. Thus, unlike the case of Riemannian geometry, but like symplectic geometry, the local theory of contact geometry is trivial, and there are no analogs of angle or curvature. However, the global theory is nontrivial, and there are globally inequivalent contact structures.

Contact form

Unlike a vector field or a covector field (i.e. a 1-form), a contact structure does not have an intrinsic sense of size or coorientation. In this sense, it can be interpreted as the space of unparameterized infinitesimal surfaces, much like how a tangent bundle can be interpreted as the space of time-parameterized infinitesimal curves.

A contact form is a 1-form that provides an intrinsic sense of size and coorientation. i.e. a smooth section of the cotangent bundle. The non-integrability condition can be given explicitly in exterior calculus:

non-zero smooth function. This makes up an ideal, called the contact ideal.

By Darboux's theorem. Such coordinates are called Darboux coordinates. In this sense, contact geometry is a stable distribution, since they are all the same up to local diffeomorphism.

is a volume form cannot be globally defined. Another obstruction is coorientability.

Coorientation

A contact structure is coorientable can be defined globally as a nonvanishing section in the cotangent bundle.

is trivial, iff the cohomology is trivial, and more specifically iff the first Stiefel–Whitney class is trivial.

Non-integrability

, the Frobenius theorem on integrability implies that the contact field ξ is completely nonintegrable. Indeed contact structures are defined as completely nonintegrable distributions. You cannot find a hypersurface in M whose tangent spaces agree with ξ, even locally. In fact, there is no submanifold of dimension greater than k whose tangent spaces lie in ξ. A submanifold that achieves this limit of dimension k is a Legendrian submanifold.

is at most order 1.

, can be thought of as a generic property is a non-generic algebraic equation. This perspective explains why it is a stable distribution.

Another perspective on non-integrability is through the Chow–Rashevskii connectivity theorem, which states that any two points in a contact manifold can be connected by a smooth curve tangent to the contact structure. This has been generalized to sub-Riemannian manifolds using the language of theoretical thermodynamics, especially Carnot cycles.

Another perspective is via the Lie algebra in the distribution such that they do not generate .

Examples

The standard contact structure

The standard contact structure in , with coordinates (x,y,z), is the one-form The contact plane ξ at a point (x,y,z) is spanned by the vectors and

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. This is an instance of the Chow–Rashevskii connectivity theorem.

. It is standard, because Darboux's theorem states that any contact structure is locally the same as the standard one.

The standard contact structure on the sphere

Given any n, the standard contact form on the (2n-1) to the unit sphere, i.e. the standard complex structure.

, which generates the Hopf fibration.

is isotropic.

One-jet

, the one-jet space is the space of germs given by the tautological 1-form.

is the graph, defined below.

Contact bundle of hyperplane elements

, its n is the bundle of its dimension-n uses projective coordinates. Any n- uniquely lifts to an n. Conversely, an n is a lift of an n, which is the standard contact structure.

is obtained by spherizing.

to be the fiber projection is spanned by translation of the base point within, as well as rotation of the hyperplane element while keeping its base point unchanged.

is an infinitesimal dimension-n. These are the points, which are infinitesimal dimension-2n necessarily has odd dimensions.

can be constructed by \R^3 at infinity is not orientable. It can be double-covered, which is coorientable..

Others

Until the 1950s, the only contact manifolds were the above ones, until Boothby and Wang in 1958 made a general construction via contactization.

The Sasakian manifolds are contact manifolds.

Brieskorn manifolds.

Every connected compact orientable three-dimensional manifold admits a contact structure. This result generalizes to any compact almost-contact manifold.

Contact transformation

A contact transformation (or contactomorphism) is a diffeomorphism between two contact manifolds that preserves their contact structure. A contact symmetry is a contact transformation from a contact manifold to itself.

then it is a strict contact transformation. Note that the concept of a strict contact transformation depends on a particular choice of contact forms, and there are inequivalent choices. Therefore, there is no "strict contact transformation" between contact structures, only between contact forms.

A strict infinitesimal contact symmetry is the Lie derivative. An infinitesimal contact symmetry on the contact manifold that generates a one-parameter family.

Examples

Projective geometry

Given a conic section in the plane, the polar reciprocation operation is an involutive. Though it exchanges points and lines, and thus is not a bijection of the plane, it exchanges line elements, thus is a bijection of the contact manifold. In words, given a line element. A curve is decomposed into a succession of tangent line elements, which are mapped to another succession of line elements. This operation exchanges envelopes and loci. In particular, if two nonlinear curves that are in contact at a point, then they remain in contact after reciprocation. This explains the name of "contact transformation".

Similarly, the line-sphere correspondence and other transformations of the Lie sphere geometry infinitesimal planes. They were in fact some of the earliest ones considered by Lie.

Legendre transformation

, then the Legendre transformation of the symplectic space. This rotation is simply multiply-by-i of the standard linear complex structure on the symplectic space. In the plane, it exchanges a curve and its dual.

itself.

. In particular, this defines the Legendre transformation for any manifold.

Canonical transformation

. This can be interpreted as a lift of the Hamilton–Jacobi equation in time-independent Hamiltonian dynamics being Hamilton's characteristic function. A canonical transformation.

Others

Given any contact form, its corresponding Reeb vector field is a strict infinitesimal contact symmetry, and the Reeb flow is a one-parameter family of contact symmetries. The codeodesic flow is one example.

For the standard contact form on an odd-dimensional sphere, its Reeb flow generates its Hopf fibration.

Submanifolds

Contact

, a contact submanifold is a contact submanifold.

Isotropic

, an isotropic submanifold (or integral submanifold.

.

Legendrian

As described above, an integral manifold can have up to n dimensions. These extremal integral manifolds are Legendrian submanifolds. Indeed, such submanifolds are extremely common, since they satisfy an h-principle:. Conversely, any Legendrian submanifold is locally of this form. Thus, each Legendrian submanifold is entirely specified by its (local) generating functions..

. Since locally, any two Legendrian n.

.

A Legendrian fibration. An equivalence of Legendre fibrations is a contactomorphism that preserves the Legendrian submanifolds too. As yet another Darboux phenomenon, any Legendrian fibration is locally the standard Legendrian fibration.

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 curves inside a contact 3-manifold. When the curve is closed, it is a Legendrian knot. Inequivalent Legendrian knots may be equivalent as smooth knots; that is, there are Legendrian knots which are smoothly isotopic to each other, but at least one intermediate knot during the isotopy must not be Legendrian. This is because Legendrian knots are rigid.

In general, 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).

Vector fields

Liouville

is (locally) Liouville. By Cartan's magic formula.

A Liouville form. The tautological 1-form is an example.

Reeb

, it has a Reeb vector field, or characteristic vector field. In particular, it shows that it is uniquely defined.

By Cartan's magic formula. Visually, if the contact form is drawn as paired hyperplane elements, then the paired hyperplane elements are preserved under Reeb vector flow.

are not parallel in general, the Reeb field is not a part of the contact structure, but rather, of the contact dynamics.

If a contact form 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.

Relation with symplectic geometry

.

Contactification

is a contact manifold.

This construction requires the cohomology class, then it can be contactified with the Boothby–Wang construction.. Take a \mathbb S^1-bundle. Any connection 1-form is the prequantization fibration.

Liouville transversal construction

is a contact manifold.

In general, define a contact type submanifold of the original symplectic manifold to be a codimension-1 submanifold that is transverse to some Liouville vector field. This construction shows that any contact type submanifold can be given a globally defined contact form, so it is coorientable.

, and suggests a reverse operation that extends a contact manifold to a symplectic manifold. Indeed, symplectization is a strict inverse to this operation in the following sense: that are symplectically isomorphic.

Reeb transversal construction

is a symplectic manifold. The Reeb vector field flow provides symplectomorphic homotopy between these, another instance of the h-principle.

Symplectization

, it can be symplectized. This is a symplectic form, as can be verified by constructing local Darboux coordinates. For example, given an n, the nonzero cotangent bundle.

There is a bijection between 1-homogeneous. This is an infinitesimal symplectomorphism that is 1-homogeneous along the fibers. Conversely, any infinitesimal symplectomorphism that is 1-homogeneous along the fibers projects down to infinitesimal contactomorphism.

generated by some 1-homogeneous Hamiltonian. This is a Lie algebra isomorphism between infinitesimal contactomorphisms and infinitesimal symplectomorphisms. This translates the well-developed theory of Hamiltonian flows to the theory of contact flows.

Contact Hamiltonian flow

Analogous to how a real-valued function (a Hamiltonian) on a symplectic manifold generates a flow, a real-valued function on a contact manifold generates a flow, which may be called a contact Hamiltonian.

.

on the submanifold.

Energy surfaces

Suppose that H is a smooth function on T*M, that E is a regular value for H, so that the level set is a smooth submanifold of codimension 1. A vector field Y is called an Euler (or Liouville) vector field if it is transverse to L and symplectic, meaning that the Lie derivative is a contact form on L.

This construction originates in Hamiltonian mechanics is Liouville.

is Liouville.

Topology

The topology of contact 3-manifolds is best understood. Given any oriented 3-manifold, there are infinitely many distinct contact structures on it. One can be constructed by performing surgery along a Legendrian link on the 3-sphere with its standard contact structure. Given any contact structure on it, applying the Lutz twist repeatedly then creates an infinite number of non-isomorphic contact structures, which are overtwisted. Any structure not overtwisted is called tight. The standard contact structure on the sphere is the only tight one possible up to isotopy.

The Giroux theorem shows that oriented contact 3-manifolds are, up to isotopy, bijective to open book decompositions up to "positive stabilization". Thus the geometry of oriented contact 3-manifolds is entirely topological.

The Weinstein conjecture is an open question asking whether on a compact contact manifold, any Reeb flow always contains a cycle. It has been proven in the 3-dimensional case.

For any geometric structure of a given type, it is important to understand whether it admits continuous deformations into non-equivalent structures. In complex structures, this is possible, leading to Teichmüller theory of Riemann surfaces and Kodaira–Spencer deformation theory. The Gray stability theorem shows that contact structures on closed manifolds cannot be deformed to a non-equivalent structure. Specifically:.The theorem does not hold for contact forms.

History

Concepts of contact geometry appear implicitly in work of Apollonius of Perga, Christiaan Huygens, Isaac Barrow, and Isaac Newton. The theory of contact transformations 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.

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 geometry has been used to describe the visual cortex.

Partial differential equations

The original motivation for the study of contact geometry was in solving first-order partial differential equations specifies a 2n-dimensional hypersurface, and the problem reduces to finding Legendrian submanifolds within this hypersurface.

Geometric optics

The Huygens–Fresnel principle of wave propagation can be formalized as a contact transformation. Specifically, given a Riemannian n-manifold, consider its unit-speed geodesic curves (i.e. parameterized by arc length). This produces a transport of unit-length tangent vectors, and thus a vector flow field on the unit tangent bundle. This is the geodesic flow. This is the cogeodesic flow. The Huygens–Fresnel principle states that the (co)geodesic flow is a strict infinitesimal contact symmetry, and more precisely, it is the Reeb vector field. This construction directly generalizes to (co)geodesic flows on Finsler manifolds.

correspond to special types of pencils of rays, and Reeb flow corresponds to ray propagation over time. That Reeb flow preserves Legendrian submanifolds implies the Malus–Dupin theorem. In particular, a single point source can be regarded either as a sphere of exiting rays, or a sphere of exiting wavefronts. They are both maximally extended compact Legendrian submanifolds. . Circular wavefronts exiting a single point in the plane is lifted to a helix. Given an involute of an evolute, the other involutes are obtained by the one-parameter family of contact transformations.

Conversely, any infinitesimal contact transformation can be cast into the form of wave propagation.

Thermodynamics

Classical thermodynamics studies systems in thermal equilibrium. Specifically, there exists a system of coordinates:

, the space of reachable states is an nInterpreted thermodynamically, the first n are the equations of state, while the last equation is the fundamental relation. Legendre transforms are special cases of contact transformations.

For any specific gas system, its reachable states is a 3-dimensional Legendrian submanifold. By changing the fundamental relation, all possible gas systems allowed by classical thermodynamics can be specified.

See also

References

Introductions to contact geometry

Applications to differential equations

Contact three-manifolds and Legendrian knots

Information on the history of contact geometry