In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, contact geometry is the study of a geometric structure on
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s given by a hyperplane
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
in the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
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
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, where one can consider either the even-dimensional
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, e.g.
geometrical optics
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of ''rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
,
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
,
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
,
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 ...
,
integrable systems
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
and to
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
. Contact geometry also has applications to
low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot th ...
; for example, it has been used by
Kronheimer
Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and former ...
and
Mrowka to prove the
property P conjecture
In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot (mathematics), knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-tri ...
, by
Michael Hutchings to define an invariant of smooth three-manifolds, and by
Lenhard Ng
Lenhard Ng (born 1976, United States) is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University.
Background and education
Lenhard Ng is an American of Chinese descent. His fath ...
to define invariants of knots. It was also used by
Yakov Eliashberg
Yakov Matveevich Eliashberg (also Yasha Eliashberg; russian: link=no, Яков Матвеевич Элиашберг; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR.
Education and career
Eliashberg receiv ...
to derive a topological characterization of
Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Stei ...
s 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
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
''M'', and a point , a contact element of ''M'' with contact point ''p'' is an (''n'' − 1)-dimensional
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to ''M'' at ''p''.
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 is any nonzero real number. Thus, the kernels of 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
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. It may be described also as the dual bundle to the tangent bundle. This may ...
T*''M'' (with the zero section
removed),
namely:
:
A contact structure on an odd dimensional manifold ''M'', of dimension , is a smooth
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
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
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
α; i.e. a smooth
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
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
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
. If ξ is co-orientable then α is defined globally.
Properties
It follows from the
Frobenius theorem on integrability 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 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.
Relation with symplectic structures
A consequence of the definition is that the restriction of the 2-form ''ω'' = ''d''α to a hyperplane in ξ is a nondegenerate 2-form. This construction provides any contact manifold ''M'' with a natural
symplectic bundle
The term "symplectic" is a calque of "complex" introduced by Hermann Weyl in 1939. In mathematics it may refer to:
* Symplectic Clifford algebra, see Weyl algebra
* Symplectic geometry
* Symplectic group
* Symplectic integrator
* Symplectic manifol ...
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
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. It may be described also as the dual bundle to the tangent bundle. This may ...
''T''*''N'' of any ''n''-dimensional manifold ''N'' is itself a manifold (of dimension 2''n'') and supports naturally an exact symplectic structure ω = ''d''λ. (This 1-form λ is sometimes called the
Liouville form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus prov ...
). There are several ways to construct an associated contact manifold, some of dimension 2''n'' − 1, some of dimension 2''n'' + 1.
;Projectivization
Let ''M'' be the
projectivization
In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multi ...
of the cotangent bundle of ''N'': thus ''M'' is fiber bundle over ''N'' whose fiber at a point ''x'' is the space of lines in T*''N'', or, equivalently, the space of hyperplanes in T''N''. 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.
;Energy surfaces
Suppose that ''H'' is a smooth function on T*''N'', 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 conformally symplectic, meaning that the Lie derivative of ''d''λ with respect to ''Y'' is a multiple of ''d''λ in a neighborhood of ''L''.
Then the restriction of
to ''L'' is a contact form on ''L''.
This construction originates in
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, 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.
;The unit cotangent bundle
Choose a
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
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 2''n''-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 ''R'', defined by the equalities
: λ(''R'') = 1 and ''d''λ(''R'', ''A'') = 0 for all vector fields ''A'',
is called the
Reeb vector field
In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifo ...
, and it generates the
geodesic flow
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection ...
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''.
;First jet bundle
On the other hand, one can build a contact manifold ''M'' of dimension 2''n'' + 1 by considering the first
jet bundle
In differential topology, the jet bundle is a certain construction that 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. Je ...
of the real valued functions on ''N''. This bundle is isomorphic to ''T''*''N''×R using the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
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''
''t''α)
is a symplectic form on ''M''×R, where ''t'' denotes the variable in the R-direction. This new manifold is called the
symplectization
In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Definition
Let (V,\xi) be a contact manifold, and let x \in V. Consider the set
: S_xV = \ \subset T^*_xV
of all nonzero 1-for ...
(sometimes
symplectification in the literature) of the contact manifold ''M''.
Examples
As a prime example, consider R
3, 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 ''x''
1, ..., ''x''
''n'', ''y''
1, ..., ''y''
''n'', one can generalize this example to any R
2''n''+1. By a
theorem of Darboux, every contact structure on a manifold looks locally like this particular contact structure on the (2''n'' + 1)-dimensional vector space.
An important class of contact manifolds is formed by
Sasakian manifold In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold (M,\theta) equipped with a special kind of Riemannian metric g, called a ''Sasakian'' metric.
Definition
A Sasakian metric is defined using the constr ...
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 (2''n'' + 1)-dimensional manifold means that no 2''n''-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 submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
s 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 knot In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into which is tangent to the standard contact structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hy ...
s 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
The term "symplectic" is a calque of "complex" introduced by Hermann Weyl in 1939. In mathematics it may refer to:
* Symplectic Clifford algebra, see Weyl algebra
* Symplectic geometry
* Symplectic group
* Symplectic integrator
* Symplectic mani ...
provides invariants of Legendrian submanifolds called
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. It is a part of a more ge ...
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
In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifo ...
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
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...
such as
symplectic field theory
The term "symplectic" is a calque of "complex" introduced by Hermann Weyl in 1939. In mathematics it may refer to:
* Symplectic Clifford algebra, see Weyl algebra
* Symplectic geometry
* Symplectic group
* Symplectic integrator
* Symplectic mani ...
and, in three dimensions,
embedded contact homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer i ...
. 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
Georges Henri Reeb (12 November 1920 – 6 November 1993) was a French mathematician. He worked in differential topology, differential geometry, differential equations, topological dynamical systems theory and non-standard analysis.
Biography
...
.
Some historical remarks
The roots of contact geometry appear in work of
Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
,
Isaac Barrow
Isaac Barrow (October 1630 – 4 May 1677) 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 proof of the fundamental theorem ...
and
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
. The theory of contact transformations (i.e. transformations preserving a contact structure) was developed by
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius Sophu ...
, with the dual aims of studying differential equations (e.g. the
Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of ...
or
canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canoni ...
) and describing the 'change of space element', familiar from
projective duality
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of du ...
.
See also
*
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...
, some flavors of which give invariants of contact manifolds and their Legendrian submanifolds
*
Quantized contact transformation
Quantization is the process of constraining an input from a continuous or otherwise large set of values (such as the real numbers) to a discrete set (such as the integers). The term ''quantization'' may refer to:
Signal processing
* Quantizatio ...
*
Sub-Riemannian geometry
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal s ...
References
Introductions to contact geometry
*
*
*
*
*
Applications to differential equations
*
Contact three-manifolds and Legendrian knots
*
Information on the history of contact geometry
*
*
*{{cite book , first=Vladimir I. , last=Arnold , title=Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals , url=https://books.google.com/books?id=7ifyBwAAQBAJ , date=2012 , orig-year=1990 , publisher=Birkhäuser , isbn=978-3-0348-9129-5
Contact geometry Theme on arxiv.org
External links
Contact manifoldat the Manifold Atlas