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.
It was also used by Yakov Eliashberg to derive a topological characterization of Stein manifolds of dimension at least six.
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 section
The 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.
It follows from the 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.
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.
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
Then 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.
