In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. From the Frobenius theorem, one recognizes that this is (roughly) the opposite of a foliation. As its sister, symplectic geometry, belongs to the even-dimensional world, in many ways contact geometry is an odd-dimensional counterpart. One difference between contact and symplectic geometry is that every 3-manifold admits a contact structure while there are cohomological obstructions to the existence of symplectic structures. For other meanings of mathematics or math, see mathematics (disambiguation). ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ... In mathematics, informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a stripy fabric. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...

Applications

Contact geometry has — as does symplectic geometry — broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, and applied mathematics such as control theory. One can prove amusing things, like 'You can always parallel-park your car, provided the space is big enough'. 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 and by Gompf to derive a topological characterization of Stein manifolds. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... See also list of optical topics. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... ‹ The template below has been proposed for deletion. ... In mathematical physics, geometric quantization is a mathematical approach to define a quantum theory corresponding to a given classical theory in such a way that certain analogies between the classical theory and the quantum theory remain manifest, for example the similarity between the Heisenberg equation in the Heisenberg picture of... In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... Parallel parking is a method of parking a vehicle in line with other parked cars. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. ... In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. ...

Contact forms and structures

A contact form α on a 2n+1 dimensional manifold M is a (local) 1-form with the property that A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

A contact structure ξ on a manifold is the kernel of a contact form α, i.e. a completely nonintegrable hyperplane field. Roughly this means that you cannot find a piece of a hypersurface tangent to ξ on an open set.

It also follows from this definition that dα, when restricted to ξ, is nondegenerate. This means that ξ is a symplectic bundle on the manifold. Since symplectic spaces are even-dimensional, contact manifolds need to be odd dimensional.

As a prime example, consider on R³, endowed with coordinates

(x, y, z),

the 1-form

dz -ydx.

The contact plane ξ at a point

(x,y,z)

is spanned by vectors

X₁ = ∂_y

and

X₂ = ∂_x+y∂_z.

(Draw a picture of this!). Actually one can generalize this example to any R²ⁿ⁺¹. By a theorem of Darboux, every contact structure on a manifold looks locally like this. This page is not about the theorem of Darboux related to the intermediate value theorem Darbouxs theorem is a theorem in symplectic topology which states that every symplectic manifold (of fixed dimension) is locally symplectomorphic. ...

The cotangent bundle T^* M of any n-dimensional manifold M is itself a manifold (of dimension 2n) and supports naturally an exact symplectic structure ω = dλ. (This 1-form λ is sometimes called Liouville form). Choose a Riemannian metric on the manifold. That allows one to consider the unit sphere in each cotangent plane. The Liouville form restricted to the unit cotangent bundle is a contact structure. The vector field A (uniquely) defined by λ(A)=1 and dλ(A, B)=0 for all vector fields B generates the geodesic flow of this metric. In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...

On the other hand, one can build a contact manifold by considering the manifold T^*M× R. With coordinates (x,t) this has a contact structure

α=dt+λ.

The last example showed how to obtain contact manifolds from symplectic ones. Vice versa one gets a symplectic manifold out of a contact manifold by crossing with R: If α is a contact form for a manifold M, then

ω=d(e^tα)

is a symplectic form on M×R, where t denotes the variable in the R-direction.

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, not 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. In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. ...

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 that can sometimes distinguish distinct Legendrian submanifolds that are topologically identical. In mathematics, Floer homology refers to a family of homology theories which share similar characteristics and are believed by experts to be closely related. ... In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. ...

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 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 such as symplectic field theory and embedded contact homology. In mathematics, Floer homology refers to a family of homology theories which share similar characteristics and are believed by experts to be closely related. ... In mathematics, Floer homology refers to a family of homology theories which share similar characteristics and are believed by experts to be closely related. ...

Some historical remarks

The roots of contact geometry appear in work of Christiaan Huygens, 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) and describing the 'change of space element', familiar from projective duality. Christiaan Huygens Christiaan Huygens (pronounced in English (IPA): ; in Dutch: )(April 14, 1629–July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ... Sir Isaac Newton, FRS (4 January 1643 – 31 March 1727) [OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, alchemist, and natural philosopher, widely regarded as one of the key figures in the history of science. ... Marius Sophus Lie (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician who largely created the theory of continuous symmetry, and applied it to the study of geometric structures and differential equations. ... In mathematics, two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: f and g are then said to be related by a Legendre transformation. ... In the geometry of projective spaces, including the projective plane, duality concerns the interchangeability between points and lines which preserves incidence properties. ...

References

Introductions to contact geometry:

• Etnyre, J. Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71 (2003), 81-107.arXiv
• Geiges, H. Contact Geometry, arXiv
• Aebischer et.al. symplectic geometry, Birkhäuser, 1994.

Contact three-manifolds and Legendrian knots:

• William Thurston, Three-Dimensional Geometry and Topology. Princeton University Press, 1997.

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.