Contact geometry

In mathematics, contact geometry is the study of a geometric structure on given by a hyperplane 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 '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.

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.

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Contact structure 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 of the tangent space to M at p. A contact structure on an odd dimensional manifold M, of dimension , is a smooth distribution of contact elements, denoted by ξ, which is Generic property (in the sense of being maximally non-integrable) at each point. A contact manifold is a smooth manifold equipped with a contact structure.

Due to the ambiguity by multiplication with a nonzero smooth function, the space of all contact elements of M can be identified with a quotient of the cotangent bundle $T^*\; M$ (with the zero section $0\_M$ removed), namely:

$\backslash text\{PT\}^*M\; =\; (\backslash text\{T\}^*M\; -\; \{0\_M\})\; /\{\backslash sim\}\; \backslash \; \backslash text\{\; where,\; for\; \}\; \backslash omega\_i\; \backslash in\; \backslash text\{T\}^*\_pM,\; \backslash \; \backslash \; \backslash omega\_1\; \backslash sim\; \backslash omega\_2\; \backslash \; \backslash iff\; \backslash \; \backslash exists\; \backslash \; \backslash lambda\; \backslash neq\; 0\; \backslash \; :\; \backslash \; \backslash omega\_1\; =\; \backslash lambda\backslash omega\_2.$

Assume that we have a smooth distribution of contact elements, ξ, given locally by a differential 1-form α; i.e. a smooth section of the cotangent bundle. The non-integrability condition can be given explicitly as:

$\backslash alpha\; \backslash wedge\; (\backslash text\{d\}\backslash alpha)^k\; \backslash neq\; 0\; \backslash \; \backslash text\{where\}\; \backslash \; (\backslash text\{d\}\backslash alpha)^k\; =\; \backslash underbrace\; \{\backslash text\{d\}\backslash alpha\; \backslash wedge\; \backslash ldots\; \backslash wedge\; \backslash text\{d\}\backslash alpha\}\_\{k\backslash text\{-times\}\}.$

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.

A contact structure is co-orientable iff there exists a global choice of the "positive" side of each contact element. That is, the contact form $\backslash alpha$ can be defined globally as a nonvanishing section in the cotangent bundle $T^*\; M$. In this case, $\backslash alpha$ is uniquely defined, up to a multiplication by a nonzero smooth function. The contact structure is not co-orientable iff nontrivial cohomology occurs, and more specifically iff the first Stiefel–Whitney class is nontrivial.

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Properties

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Integrability Because $\backslash alpha\; \backslash wedge\; (\backslash text\{d\}\backslash alpha)^k\; \backslash neq\; 0$, the Frobenius theorem on integrability implies that the contact field ξ is completely nonintegrable. Conversely, 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. A submanifold that achieves this limit of dimension k is a Legendrian submanifold.

Maximal non-integrability, as defined by $\backslash alpha\; \backslash wedge\; (\backslash text\{d\}\backslash alpha)^k\; \backslash neq\; 0$, can be thought of as a generic property of distributions, since $\backslash alpha\; \backslash wedge\; (\backslash text\{d\}\backslash alpha)^k\; =\; 0$ is an algebraic condition on the derivatives of the components of $\backslash alpha$, which is not generic.

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Examples

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The standard contact structure standard contact structure in R³, with coordinates ( x, y, z), is the one-form The contact plane ξ at a point ( x, y, z) is spanned by the vectors and

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.

This example generalizes to any $\backslash R^\{2n+1\}$. Its standard contact structure is $\backslash theta\; :=\; dz\; -\; \backslash sum\_\{i=1\}^n\; y\_i\; dx\_i$. By a theorem of Darboux, every contact structure on a manifold is locally $\backslash theta$, in the sense that around each point, there is a local choice of coordinates in which $\backslash theta\; =\; dz\; -\; \backslash sum\_\{i=1\}^n\; y\_i\; dx\_i$.

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Others The Sasakian manifolds are contact manifolds.

Every Connected space Compact space Orientability three-dimensional manifold admits a contact structure.

This result generalizes to any compact almost-contact manifold.

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Legendrian submanifolds 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 ( n + 1)-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 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 curves inside a contact 3-manifold. When the curve is closed, it is a Legendrian knot. Inequivalent Legendrian knots may be equivalent as smooth knots; that is, there are Legendrian knots which are smoothly isotopic to each other, but at least one intermediate knot during the isotopy must not be Legendrian. This is because Legendrian knots are rigid.

In general, Legendrian submanifolds are very rigid objects; typically there are infinitely many Legendrian isotopy classes of embeddings which are all smoothly isotopic. 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).

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Relation with symplectic geometry 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.

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Projectivization Let M be the projective space 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.

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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 $L=\backslash \{(q,p)\backslash in\; T^*N\backslash mid\; H(q,p)=E\backslash \}$ 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 $i\_Y\; \backslash ,d\backslash lambda$ 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.

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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 qs 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, 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.

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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 ^tα)

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.

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Reeb vector field Given a contact manifold $(M,\; \backslash alpha)$, it has a Reeb vector field, or characteristic vector field $R$. It is uniquely defined by $$d\backslash alpha(R,\; \backslash cdot)\; =\; 0,\; \backslash ;\; \backslash alpha(R)\; =\; 1$$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 symplectic field theory and, in three dimensions, 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.

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History 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.

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Applications Like symplectic geometry, contact geometry has broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, 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 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. It was also used by Yakov Eliashberg to derive a topological characterization of of dimension at least six.

Contact geometry has been used to describe the visual cortex.

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See also

• Floer homology, some flavors of which give invariants of contact manifolds and their Legendrian submanifolds
• Sub-Riemannian geometry

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Introductions to contact geometry

• J. Etnyre (2025). 9780821835074 ISBN 9780821835074
• Hansjörg Geiges (2025). 9781139467957, Cambridge University Press. . ISBN 9781139467957
• Aebischer (1994). 9783764350642, Birkhäuser. ISBN 9783764350642
• V.I. Arnold (1989). 9780387968902, Springer-Verlag. ISBN 9780387968902

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Applications to differential equations

• V.I. Arnold (1988). 9780387966496, Springer-Verlag. ISBN 9780387966496

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Contact three-manifolds and Legendrian knots

• William Thurston (1997). 9780691083049, Princeton University Press. ISBN 9780691083049

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Information on the history of contact geometry

• Vladimir I. Arnold (2025). 9783034891295, Birkhäuser. . ISBN 9783034891295
• Contact geometry Theme on arxiv.org

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External links

• Contact manifold at the Manifold Atlas

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