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'.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.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.[2][3] 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.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 sectionThe non-integrability condition can be given explicitly as:[2] 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.This property of the contact field is roughly the opposite of being a field formed from the tangent planes of a family of nonoverlapping hypersurfaces in M. In particular, you cannot find a hypersurface in M whose tangent spaces agree with ξ, even locally.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 dλ with respect to Y is a multiple of dλ in a neighborhood of L. Then the restriction ofThen the Liouville form restricted to the unit cotangent bundle is a contact structure.Every connected compact orientable three-dimensional manifold admits a contact structure.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.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.
The standard contact structure on R 3 . Each point in R 3 has a plane associated to it by the contact structure, in this case as the kernel of the one-form d z − y d x . 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.
mathematicssmooth manifoldsdistributiontangent bundlecomplete integrabilityfoliationFrobenius theoremsymplectic geometryclassical mechanicsphase spacephysicsgeometrical opticsthermodynamicsgeometric quantizationcontrol theorylow-dimensional topologyKronheimerMrowkaproperty P conjectureMichael HutchingsLenhard NgYakov EliashbergStein manifoldsvisual cortexsmooth manifoldlinear subspacetangent spacecotangent bundledifferential 1-formsectionsmooth functionFrobenius theorem on integrabilityLiouville formprojectivizationHamiltonian mechanicsRiemannian metricvector fieldgeodesic flowjet bundleexterior derivativesymplectizationtheorem of DarbouxSasakian manifoldsconnectedcompactorientablealmost-contact manifoldLagrangian submanifoldsLegendrian knotsrelative contact homologyReeb vector fieldFloer homologyGeorges ReebChristiaan HuygensIsaac BarrowIsaac NewtonSophus LieLegendre transformationcanonical transformationprojective dualitySub-Riemannian geometryThurston, William