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'. Equivalently, such a distribution may be given 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.

Mathematical formulation

Contact structure

Given an -dimensional smooth manifold, and a point, a contact element of with contact point is an -dimensional linear subspace of the tangent space to at. A contact structure on an odd dimensional manifold, of dimension, is a smooth distribution of contact elements, denoted by, which is generic 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 can be identified with a quotient of the cotangent bundle , namely:
for, with.
Equivalently, a contact structure can be defined as a completely non-integrable section of, the -th contact bundle of.
By Darboux's theorem, all contact structures of the same dimension are locally diffeomorphic. Thus, unlike the case of Riemannian geometry, but like symplectic geometry, the local theory of contact geometry is trivial, and there are no analogs of angle or curvature. However, the global theory is nontrivial, and there are globally inequivalent contact structures.

Contact form

Unlike a vector field or a covector field, a contact structure does not have an intrinsic sense of size or coorientation. In this sense, it can be interpreted as the space of unparameterized infinitesimal surfaces, much like how a tangent bundle can be interpreted as the space of time-parameterized infinitesimal curves.
A contact form is a 1-form that provides an intrinsic sense of size and coorientation. i.e. a smooth section of the cotangent bundle. The non-integrability condition can be given explicitly in exterior calculus:
Note that given any non-zero smooth function, gives the same contact structure. In order to absorb the ambiguity of magnitude, one can consider the set of all for an arbitrary smooth. This makes up an ideal of all 1-forms on, called the contact ideal.
By Darboux's theorem, around any point there is a neighborhood with a coordinate system, such that. Such coordinates are called Darboux coordinates. In this sense, contact geometry is a stable distribution, since they are all the same up to local diffeomorphism.
does not need to be globally defined. Indeed sometimes it cannot be globally defined due to topological obstructions. One obstruction is that if is globally defined, then is a volume form, thus is orientable. Thus if is not orientable, then cannot be globally defined. Another obstruction is coorientability.

Coorientation

A contact structure is coorientable iff there exists a global choice of the "positive" side of each contact element. That is, the contact form can be defined globally as a nonvanishing section in the cotangent bundle. In this case, is uniquely defined, up to a multiplication by a nonzero smooth function. A coorientation can be defined as a global nonzero section of the line bundle.
The contact structure is coorientable iff is trivial, iff the cohomology is trivial, and more specifically iff the first Stiefel–Whitney class is trivial.

Non-integrability

Because, the Frobenius theorem on integrability implies that the contact field ξ is completely nonintegrable. Indeed contact structures are defined as completely nonintegrable distributions. 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.
For 3-manifolds, there is a geometric characterization of contact structures on it. A distribution of plane elements in a 3-manifold is a contact structure iff on any point on any embedded surface, the contact at between and is at most order 1.
Maximal non-integrability, as defined by, can be thought of as a generic property of distributions, since is a non-generic algebraic equation on the derivatives of the components of. This perspective explains why it is a stable distribution.
Another perspective on non-integrability is through the Chow–Rashevskii connectivity theorem, which states that any two points in a contact manifold can be connected by a smooth curve tangent to the contact structure. This has been generalized to sub-Riemannian manifolds using the language of theoretical thermodynamics, especially Carnot cycles.
Another perspective is via the Lie algebra of the distribution. There exists up to vector fields in the distribution such that they do not generate.

Examples

The standard contact structure

The standard contact structure in , with coordinates, is the one-form The contact plane ξ at a point 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 is an instance of the Chow–Rashevskii connectivity theorem.
This example generalizes to any. Its standard contact structure is. It is standard, because Darboux's theorem states that any contact structure is locally the same as the standard one.

The standard contact structure on the sphere

Given any n, the standard contact form on the -sphere is obtained by restricting the Liouville 1-form on to the unit sphere. Equivalently, it is obtained by the Liouville 1-form on , where is the multiplication by, i.e. the standard complex structure on.
The Reeb vector field is, which generates the Hopf fibration.
Equivalently, consider the standard symplectic structure on. Each 1-dimensional subspace is isotropic, and has a complementary coisotropic subspace that contains it. Projectivized to, each point in has a complementary plane that contains the point. This distribution of planes is isomorphic to the standard contact structure on.

One-jet

Given a manifold of dimension, the one-jet space is the space of germs of type identified up to order-1 contact. Intuitively, each point in is a mapping from an infinitesimal neighborhood of to. Each member of the space can be identified by the three quantities , thus is a manifold of dimension and can be identified with. It has a natural contact form given by the tautological 1-form. The standard contact structure is the special case where.
Any first-differentiable function then uniquely lifts to a Legendrian submanifold in, and conversely, any Legendrian submanifold is the lift of a first-differentiable function. Its projection to is the graph of the function. This also shows that embeds into the contact bundle of hyperplane elements, defined below.

Contact bundle of hyperplane elements

Given a manifold of dimension, its n-th contact bundle is the bundle of its dimension-n contact elements. More abstractly, it is the projectivized cotangent bundle. Locally, expand in coordinates as, then the contact bundle locally has coordinates, where uses projective coordinates. Any n-submanifold of uniquely lifts to an n-submanifold of. Conversely, an n-submanifold of is a lift of an n-submanifold of iff it annihilates the 1-form. On the subset where, the condition becomes, which is the standard contact structure.
Similarly, the contact bundle of cooriented hyperplane elements is obtained by spherizing the cotangent bundle, i.e. quotienting only by.
The contact structure on can also be described coordinate-free. Define to be the fiber projection that maps a hyperplane element to its base point. Then, for any, a local tangent vector is a simultaneous translation of the base point and a rotation of the hyperplane element. Then is in the hyper-hyperplane at iff is in the hyperplane element of itself. In other words, the -dimensional hyper-hyperplane at is spanned by translation of the base point within, as well as rotation of the hyperplane element while keeping its base point unchanged.
Be careful with two meanings of hyperplanes here. A hyperplane element on is an infinitesimal dimension-n hyperplane in. These are the points of the contact manifold. The contact structure of consists of hyperplane elements in, which are infinitesimal dimension-2n hyperplanes in. The contact structure is not over, which can have even dimensions, whereas necessarily has odd dimensions.
When, is the contact bundle of line elements in the plane, and is homeomorphic to the direct product of the plane with the projective 1-space. The contact structure of looks like plane elements that rotate around their axis as they move along the "vertical" direction, completing a 180° when it finishes one cycle through. The standard contact structure in can then be induced via a map. Equivalently, the contact structure on can be constructed by gluing at infinity. However, whereas the contact structure on is coorientable, that on is not, since of is not orientable. It can be double-covered by, which is coorientable. A circle in the plane lifts to a helix in, but a double helix in.