In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution

Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties...

 in the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 and specified by a one-form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem

Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...

, one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one foliation

Foliation

In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....

 on the manifold ('complete integrability').

Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, which belongs to the even-dimensional world. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, where one can consider either the even-dimensional phase space

Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

 of a mechanical system or the odd-dimensional extended phase space that includes the time variable.

Applications

Contact geometry has — as does symplectic geometry — broad applications in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, e.g. geometrical optics

Geometrical optics

Geometrical optics, or ray optics, describes light propagation in terms of "rays". The "ray" in geometric optics is an abstraction, or "instrument", which can be used to approximately model how light will propagate. Light rays are defined to propagate in a rectilinear path as far as they travel in...

, classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, thermodynamics

Thermodynamics

Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, geometric quantization

Geometric quantization

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory...

, and applied mathematics such as control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

. One can prove amusing things, like 'You can always parallel-park

Parallel parking

thumb|250px|right|Parallel-parked cars in [[Washington, D.C.]]thumb|250px|right|A motorist gets assistance parallel-parkingParallel parking is a method of parking a vehicle in line with other parked cars. Cars parked in parallel are in one line, parallel to the curb, with the front bumper of each...

 your car, provided the space is big enough'.
Contact geometry also has applications to low-dimensional topology

Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...

; for example, it has been used by Kronheimer and Mrowka

Tomasz Mrowka

Tomasz Mrowka is a Polish American mathematician. He has been the Singer Professor of Mathematics at Massachusetts Institute of Technology since 2010. A graduate of MIT, he received the Ph.D. from University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby...

 to prove the property P conjecture

Property P conjecture

In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing Dehn surgery on the knot is non-simply-connected...

 and by Gompf to derive a topological characterization of Stein manifold

Stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...

s.

Contact forms and structures

Given an n-dimensional smooth manifold M, and a point , a contact element of M with contact point p is an (n − 1)-dimensional linear subspace

Linear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

 of the tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

 to M at p. A contact element can be given by the zeros of a 1-form on the tangent space to M at p. However, if a contact element is given by the zeros of a 1-form ω, then it will also be given by the zeros of λω where . Thus, 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

Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...

 PT*M, where:

A contact structure on an odd dimensional manifold M, of dimension , is a smooth distribution of contact elements, denoted by ξ, which is generic at each point. 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

Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...

 of the cotangent bundle. The non-integrability condition can be given explicitly as:
Notice that if ξ is given by the differential 1-form α, then the same distribution is given locally by , where ƒ is a non-zero smooth function

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

. If ξ is co-orientable then α is defined globally.

Properties

It follows from the Frobenius theorem on integrability

Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...

 that the contact field ξ is completely nonintegrable. This property of the contact field is roughly the opposite of being a field formed by the tangent planes to a family of nonoverlapping hypersurfaces in M. In particular, you cannot find a piece of a hypersurface tangent to ξ on an open set of M. More precisely, a maximally integrable subbundle has dimension n.

Relation with symplectic structures

A consequence of the definition is that the restriction of the 2-form ω = dα to a hyperplane in ξ is a nondegenerate 2-form. This construction provides any contact manifold M with a natural symplectic bundle of rank one smaller than the dimension of M. Note that a symplectic vector space is always even-dimensional, while contact manifolds need to be odd-dimensional.

The cotangent bundle

Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...

 T*N of any n-dimensional manifold N is itself a manifold (of dimension 2n) and supports naturally an exact symplectic structure ω = dλ. (This 1-form λ is sometimes called the Liouville form). There are several ways to construct an associated contact manifold, one of dimension 2n − 1, one of dimension 2n + 1.

1. Let M be the projectivization
   Projective space
   In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
   
    of the cotangent bundle of N: thus M is fiber bundle over a M whose fiber at a point x is the space of lines in T*N, or, equivalently, the space of hyperplanes in TN. The 1-form λ does not descend to a genuine 1-form on M. However, it is homogeneous of degree 1, and so it defines a 1-form with values in the line bundle O(1), which is the dual of the fibrewise tautological line bundle of M. The kernel of this 1-form defines a contact distribution.
2. Suppose that H is a smooth function on T*N, that E is a regular value for H, and that there is a vector field Y

-- called an `Euler' or `Liousville' vector field --transverse to the level set of H
, and conformally symplectic, meaning that the Lie derivative
of dλ with respect to Y is again dλ -at least in a neighborhood of this level set.
Then the restriction of dλ(Y, *) to the level set is a contact form on the level set.
This construction originates in Hamiltonian mechanics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

, where H is a Hamiltonian of a mechanical system with the configuration space N and the phase space T*N, and E is the value of the energy.

1. Choose a Riemannian metric on the manifold N and let H be the associated kinetic energy.

Then the level set H =1/2 is the unit cotangent bundle of N, a smooth manifold of dimension 2n-1 fibering over
N with fibers being spheres. Then the Liouville form restricted to the unit cotangent bundle is a contact structure. This corresponds to a special case of the second construction, where the flow of the
Euler vector field Y corresponds to linear scaling of momenta p's, leaving the q's fixed.
The vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 R, defined by the equalities

1. λ(R) = 1 and dλ(R, A) = 0 for all vector fields A,

   is called the Reeb vector field, and it generates the geodesic flow of the Riemannian metric. More precisely, using the Riemannian metric, one can identify each point of the cotangent bundle of N with a point of the tangent bundle of N, and then the value of R at that point of the (unit) cotangent bundle is the corresponding (unit) vector parallel to N.

2. On the other hand, one can build a contact manifold M of dimension 2n + 1 by considering the first jet bundle
   Jet bundle
   In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...
   
    of the real valued functions on N. This bundle is isomorphic to T*N×R using the exterior derivative
   Exterior derivative
   In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
   
    of a function. With coordinates (x, t), M has a contact structure

   α = dt + λ.

Conversely, given any contact manifold M, the product M×R has a natural structure of a symplectic manifold. If α is a contact form on M, then

ω = d(e^tα)

is a symplectic form on M×R, where t denotes the variable in the R-direction. This new manifold is called the symplectization

Symplectization

In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.- Definition :Let be a contact manifold, and let x \in V. Consider the setof all nonzero 1-forms at x, which have the contact plane \xi_x as their kernel...

 (sometimes symplectification in the literature) of the contact manifold M.

Examples

As a prime example, consider R³, endowed with coordinates (x,y,z) and the one-form The contact plane ξ at a point (x,y,z) is spanned by the vectors and
By replacing the single variables x and y with the multivariables x₁, ..., x_n, y₁, ..., y_n, one can generalize this example to any R²ⁿ⁺¹. By a theorem of Darboux, every contact structure on a manifold looks locally like this particular contact structure on the (2n + 1)-dimensional vector space.

An important class of contact manifolds is formed by Sasakian manifold

Sasakian manifold

In differential geometry, a Sasakian manifold is a contact manifold equipped with a special kind of Riemannian metric g, called a Sasakian metric.-Definition:...

s.

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. 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.

Legendrian submanifolds are very rigid objects; in some situations, being Legendrian forces submanifolds to be unknotted. Symplectic field theory provides invariants of Legendrian submanifolds called relative contact homology

Relative contact homology

In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its Legendrian submanifolds...

 that can sometimes distinguish distinct Legendrian submanifolds that are topologically identical.

Reeb vector field

If α is a contact form for a given contact structure, the Reeb

Georges Reeb

Georges Henri Reeb was a French mathematician. He worked in differential topology, differential geometry, differential equations, topological dynamical systems theory and non-standard analysis....

 vector field R can be defined as the unique element of the kernel of dα such that α(R) = 1. Its dynamics can be used to study the structure of the contact manifold or even the underlying manifold using techniques of Floer homology

Floer homology

Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...

 such as symplectic field theory and embedded contact homology.

Some historical remarks

The roots of contact geometry appear in work of Christiaan Huygens, Isaac Barrow

Isaac Barrow

Isaac Barrow was an English Christian theologian, and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was...

 and Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

. The theory of contact transformations (i.e. transformations preserving a contact structure) was developed by Sophus Lie

Sophus Lie

Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...

, with the dual aims of studying differential equations (e.g. the Legendre transformation

Legendre transformation

In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another...

 or canonical transformation

Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates  →  that preserves the form of Hamilton's equations , although it...

) and describing the 'change of space element', familiar from projective duality.

Introductions to contact geometry

• Etnyre, J. Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71 (2003), 81–107, math.SG/0111118
• Geiges, H. Contact Geometry, math.SG/0307242
• Geiges, H. An Introduction to Contact Topology, Cambridge University Press, 2008.
• Aebischer et al. Symplectic geometry, Birkhäuser (1994), ISBN 3-7643-5064-4
• V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag (1989), ISBN 0-387-96890-3

Applications to differential equations

• V. I. Arnold, Geometrical Methods In The Theory Of Ordinary Differential Equations, Springer-Verlag (1988), ISBN 0-387-96649-8

Contact three-manifolds and Legendrian knots

• William Thurston, Three-Dimensional Geometry and Topology. Princeton University Press(1997), ISBN 0-691-08304-5

Information on the history of contact geometry

• Lutz, R. Quelques remarques historiques et prospectives sur la géométrie de contact , Conf. on Diff. Geom. and Top. (Sardinia, 1988) Rend. Fac. Sci. Univ. Cagliari 58 (1988), suppl., 361–393.
• Geiges, H. A Brief History of Contact Geometry and Topology, Expo. Math. 19 (2001), 25–53.
• Arnold, V.I. (trans. E. Primrose), Huygens and Barrow, Newton and Hooke: pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals. Birkhauser Verlag, 1990.
• Contact geometry Theme on arxiv.org