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Contact geometry
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{{Short description|Branch of geometry}} {{redirect|Contact form|web email forms|Form (web)#Form-to-email scripts}} [[File:Standard contact structure.svg|thumb|right|The standard contact structure on '''R'''<sup>3</sup>. Each point in '''R'''<sup>3</sup> has a plane associated to it by the contact structure, in this case as the kernel of the one-form {{nowrap|1=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.]] In [[mathematics]], '''contact geometry''' is the study of a geometric structure on [[smooth manifold]]s given by a hyperplane [[distribution (differential geometry)|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 '[[integrable system|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 (differential topology)|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. ==Applications== Like symplectic geometry, contact geometry has broad applications in [[physics]], e.g. [[geometrical optics]], [[classical mechanics]], [[thermodynamics]], [[geometric quantization]], [[Dispersionless equation#Multidimensional integrable dispersionless systems|integrable systems]] and to [[control theory]]. Contact geometry also has applications to [[low-dimensional topology]]; for example, it has been used by [[Kronheimer]] and [[Tomasz Mrowka|Mrowka]] to prove the [[property P conjecture]], by [[Michael Hutchings (mathematician)|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 manifold]]s of dimension at least six. Contact geometry has been used to describe the [[visual cortex]].<ref>{{Cite journal |last=Hoffman |first=William C. |date=1989-08-01 |title=The visual cortex is a contact bundle |url=https://www.sciencedirect.com/science/article/pii/009630038990091X |journal=Applied Mathematics and Computation |volume=32 |issue=2 |pages=137–167 |doi=10.1016/0096-3003(89)90091-X |issn=0096-3003|url-access=subscription }}</ref> ==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]] ''M'', and a point {{nowrap|1=''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''.<ref name="MMCM">{{Citation |first=V.I. |last=Arnold |chapter=Appendix 4 Contact structures |title=Mathematical Methods of Classical Mechanics|publisher=Springer|year=1989 |isbn=0-387-96890-3 |chapter-url=https://archive.org/details/mathematicalmeth0000arno/page/349 |pages=349−370 }}</ref><ref name="CGWP">{{cite journal|last=Arnold|first=V.I.|year=1989|title=Contact Geometry and Wave Propagation|journal=Monographie de l'Enseignement Mathématique |series=Conférences de l'Union Mathématique Internationale |publisher=Université de Genève |zbl=0694.53001 |issn=0425-0818 }}</ref> 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 {{nowrap|1=λ ≠ 0}} is any nonzero real number. Thus, the kernels of {{nowrap|1={ λω : λ ≠ 0 } }} 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]] T*''M'' (with the zero section <math> 0_M </math> removed),<ref name="MMCM"/> namely: :<math>\text{PT}^*M = (\text{T}^*M - {0_M}) /{\sim} \ \text{ where, for } \omega_i \in \text{T}^*_pM, \ \ \omega_1 \sim \omega_2 \ \iff \ \exists \ \lambda \neq 0 \ : \ \omega_1 = \lambda\omega_2.</math> A '''contact structure''' on an odd dimensional manifold ''M'', of dimension {{nowrap|1=2''k'' + 1}}, is a smooth [[Distribution_(differential_geometry)|distribution]] of contact elements, denoted by ''ξ'', which is generic at each point.<ref name="MMCM"/><ref name="CGWP"/> The genericity condition is that ''ξ'' is [[Integrable_system#Frobenius_integrability_.28overdetermined_differential_systems.29|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 (fiber bundle)|section]] of the cotangent bundle. The non-integrability condition can be given explicitly as:<ref name="MMCM"/> :<math> \alpha \wedge (\text{d}\alpha)^k \neq 0 \ \text{where} \ (\text{d}\alpha)^k = \underbrace {\text{d}\alpha \wedge \ldots \wedge \text{d}\alpha}_{k\text{-times}}.</math> Notice that if ''ξ'' is given by the differential 1-form ''α'', then the same distribution is given locally by {{nowrap|1=''β'' = ƒ⋅''α''}}, where ƒ is a non-zero [[smooth function]]. If ''ξ'' is co-orientable then ''α'' is defined globally. ===Properties=== It follows from the [[Frobenius theorem (differential topology)|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 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. 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'' = ''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]] ''T''*''N'' of any ''n''-dimensional manifold ''N'' is itself a manifold (of dimension 2''n'') 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, some of dimension 2''n'' − 1, some of dimension 2''n'' + 1. ;Projectivization Let ''M'' be the [[projective space|projectivization]] 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 <math>L=\{(q,p)\in T^*N\mid H(q,p)=E\}</math> 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 ''dλ'' with respect to ''Y'' is a multiple of ''dλ'' in a neighborhood of ''L''. Then the restriction of <math>i_Y \,d\lambda</math> to ''L'' is a contact form on ''L''. This construction originates in [[Hamiltonian mechanics]], 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]] 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 ''dλ''(''R'', ''A'') = 0 for all vector fields ''A'', is called the '''[[#Reeb vector field|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''. ;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]] of the real valued functions on ''N''. This bundle is isomorphic to ''T''*''N''×'''R''' using the [[exterior derivative]] 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''<sup>''t''</sup>α) is a symplectic form on ''M''×'''R''', where ''t'' denotes the variable in the '''R'''-direction. This new manifold is called the [[symplectization]] (sometimes [[symplectification]] in the literature) of the contact manifold ''M''. ===Examples=== As a prime example, consider '''R'''<sup>3</sup>, endowed with coordinates (''x'',''y'',''z'') and the one-form {{nowrap|1=''dz'' − ''y'' ''dx''.}} The contact plane ''ξ'' at a point (''x'',''y'',''z'') is spanned by the vectors {{nowrap|1=''X''<sub>1</sub> = <big>∂</big><sub>''y''</sub>}} and {{nowrap|1=''X''<sub>2</sub> = <big>∂</big><sub>''x''</sub> + ''y'' <big>∂</big><sub>''z''</sub>.}} <!--- Presumably, a challenge to editors to insert a picture of a contact manifold? (Draw a picture of this!). ---> By replacing the single variables ''x'' and ''y'' with the multivariables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, ''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub>, one can generalize this example to any '''R'''<sup>2''n''+1</sup>. By a [[Darboux theorem|theorem of Darboux]], every contact structure on a manifold looks locally like this particular contact structure on the (2''n'' + 1)-dimensional vector space. The [[Sasakian manifold|Sasakian manifolds]] comprise an important class of contact manifolds. Every [[Connected space|connected]] [[Compact space|compact]] [[Orientability|orientable]] three-dimensional manifold admits a contact structure.<ref>{{Cite journal |last=Martinet |first=J. |date=1971 |editor-last=Wall |editor-first=C. T. C. |title=Formes de Contact sur les Variétés de Dimension 3 |url=https://link.springer.com/chapter/10.1007/BFb0068901 |journal=Proceedings of Liverpool Singularities Symposium II |language=fr |location=Berlin, Heidelberg |publisher=Springer |pages=142–163 |doi=10.1007/BFb0068901 |isbn=978-3-540-36868-7|url-access=subscription }}</ref> This result generalises to any compact [[almost-contact manifold]].<ref>{{Cite journal |last=Borman |first=Matthew Strom |last2=Eliashberg |first2=Yakov |last3=Murphy |first3=Emmy |date=2015 |title=Existence and classification of overtwisted contact structures in all dimensions |url=https://projecteuclid.org/journals/acta-mathematica/volume-215/issue-2/Existence-and-classification-of-overtwisted-contact-structures-in-all-dimensions/10.1007/s11511-016-0134-4.full |journal=Acta Mathematica |volume=215 |issue=2 |pages=281–361 |doi=10.1007/s11511-016-0134-4 |issn=0001-5962|arxiv=1404.6157 }}</ref> ==Legendrian submanifolds and knots==<!-- This section is linked from [[Knot theory]] --> 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]]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]]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. [[Floer homology#Symplectic field theory (SFT)|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). ==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 (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]] such as [[Floer homology#Symplectic field theory (SFT)|symplectic field theory]] and, in three dimensions, [[Floer homology#Embedded contact homology|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]]. ==Some historical remarks== The roots of contact geometry appear in work of [[Christiaan Huygens]], [[Isaac 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]] 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<ref>{{Cite journal |last=Boothby |first=W. M. |last2=Wang |first2=H. C. |date=1958 |title=On Contact Manifolds |url=https://www.jstor.org/stable/1970165 |journal=Annals of Mathematics |volume=68 |issue=3 |pages=721–734 |doi=10.2307/1970165 |issn=0003-486X|url-access=subscription }}</ref><ref>{{Cite journal |last=Geiges |first=Hansjörg |date=2001-01-01 |title=A brief history of contact geometry and topology |url=https://www.sciencedirect.com/science/article/pii/S0723086901800141 |journal=Expositiones Mathematicae |volume=19 |issue=1 |pages=25–53 |doi=10.1016/S0723-0869(01)80014-1 |issn=0723-0869|doi-access=free }}</ref><ref>{{Cite web |last=Sloman |first=Leila |date=2023-11-07 |title=In the ‘Wild West’ of Geometry, Mathematicians Redefine the Sphere |url=https://www.quantamagazine.org/in-the-wild-west-of-geometry-mathematicians-redefine-the-sphere-20231107/ |access-date=2023-11-07 |website=Quanta Magazine |language=en}}</ref> ==See also== *[[Floer homology]], some flavors of which give invariants of contact manifolds and their Legendrian submanifolds *[[Sub-Riemannian geometry]] ==References== {{reflist}} ===Introductions to contact geometry=== *{{cite journal |first=J. |last=Etnyre |title=Introductory lectures on contact geometry |journal=Proc. Sympos. Pure Math. |series=Proceedings of Symposia in Pure Mathematics |volume=71 |issue= |pages=81–107 |year=2003 |doi= 10.1090/pspum/071/2024631|arxiv=math/0111118|isbn=9780821835074 |s2cid=6174175 }} *{{cite arXiv |last=Geiges |first=H. |eprint=math/0307242 |title=Contact Geometry|year=2003 }} *{{cite book |first=Hansjörg |last=Geiges |title=An Introduction to Contact Topology |url=https://books.google.com/books?id=RERR4zMDYRgC |date=2008 |publisher=Cambridge University Press |isbn=978-1-139-46795-7}} *{{cite book |last=Aebischer |title=Symplectic geometry |publisher=Birkhäuser |year=1994 |isbn=3-7643-5064-4 }} *{{cite book |first=V.I. |last=Arnold |title=Mathematical Methods of Classical Mechanics |publisher=Springer-Verlag |year=1989 |isbn=0-387-96890-3 }} ===Applications to differential equations=== *{{cite book |first=V.I. |last=Arnold |title=Geometrical Methods In The Theory Of Ordinary Differential Equations |publisher=Springer-Verlag |year=1988 |isbn=0-387-96649-8 }} ===Contact three-manifolds and Legendrian knots=== *{{cite book |author-link=William Thurston |first=William |last=Thurston |title=Three-Dimensional Geometry and Topology |publisher=Princeton University Press |year=1997 |isbn=0-691-08304-5 }} ===Information on the history of contact geometry=== *{{cite book |first=R. |last=Lutz |chapter=Quelques remarques historiques et prospectives sur la géométrie de contact |title=Conference on Differential Geometry and Topology (Sardinia, 1988) |series=Rend. Fac. Sci. Univ. Cagliari |year=1988 |volume=58 suppl |pages=361–393 |mr=1122864}} *{{cite journal |first=H. |last=Geiges |title=A Brief History of Contact Geometry and Topology |journal=Expo. Math. |volume=19 |issue= |pages=25–53 |year=2001 |doi=10.1016/S0723-0869(01)80014-1 |doi-access=free }} *{{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}} *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-265776 Contact geometry Theme on arxiv.org] ==External links== *[http://www.map.mpim-bonn.mpg.de/Contact_manifold Contact manifold] at the Manifold Atlas [[Category:Contact geometry| ]]
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