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 (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.

Given an ${\displaystyle n}$-dimensional smooth manifold ${\displaystyle M}$, and a point ${\displaystyle p\in M}$, a contact element of ${\displaystyle M}$ with contact point ${\displaystyle p}$ is an ${\displaystyle n-1}$-dimensional linear subspace of the tangent space to ${\displaystyle M}$ at ${\displaystyle p}$. A contact structure on an odd dimensional manifold ${\displaystyle M}$, of dimension ${\displaystyle 2n+1}$, is a smooth distribution of contact elements, denoted by ${\displaystyle \xi }$, which is generic (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 ${\displaystyle M}$ can be identified with a quotient of the cotangent bundle ${\displaystyle T^{*}M}$ (with the zero section ${\displaystyle 0_{M}}$ removed), namely: $${\displaystyle \mathrm {PT} ^{*}M=(T^{*}M\setminus 0_{M})/{\sim }}$$ for ${\displaystyle \omega _{i}\in T_{p}^{*}M}$, ${\displaystyle \omega _{1}\sim \omega _{2}\iff \exists \,\lambda \neq 0}$ with ${\displaystyle \omega _{1}=\lambda \omega _{2}}$.

Equivalently, a contact structure can be defined as a completely non-integrable section of ${\displaystyle C_{2n}M}$, the ${\displaystyle 2n}$-th contact bundle of ${\displaystyle M}$.

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.

Unlike a vector field or a covector field (i.e. a 1-form), 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 ${\displaystyle \alpha }$ 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: