Kontaktgeometri

Kontaktgeometri, or contact geometry, is a branch of differential geometry that studies contact manifolds—odd-dimensional manifolds equipped with a maximally non-integrable hyperplane field. A contact structure on a (2n+1)-dimensional manifold M is given by a 1-form α such that ξ = ker α is a hyperplane distribution and α ∧ (dα)^n is nowhere vanishing. The form α is called a contact form, and changing α by a nonzero function yields the same underlying contact geometry up to contactomorphisms.

A fundamental result is Darboux’s theorem for contact geometry: every point has a neighborhood in which one can find coordinates (x1, ..., xn, y1, ..., yn, z) with α = dz − ∑ y_i dx_i, so locally all contact structures look the same. The geometry is often studied up to contactomorphisms, diffeomorphisms that pull back a contact form to a nonzero multiple of itself.

Associated objects include the Reeb vector field R, uniquely defined by α(R) = 1 and i_R dα = 0, and submanifolds called Legendrian submanifolds, which are n-dimensional and everywhere tangent to ξ. In three-dimensional space with the standard structure, Legendrian knots are curves tangent to the contact planes.

Key examples include the standard contact structure on R^{2n+1} with α = dz − ∑ y_i dx_i, and the canonical contact structure on the 1-jet bundle J^1(M) of a manifold M. Connections exist to symplectic geometry via symplectization: the product M × R with the symplectic form d(e^t α) yields a bridge to even-dimensional geometry.

Historically, contact geometry arose from 19th-century studies of contact transformations by Sophus Lie and was developed further by Élie Cartan, among others. It remains a central framework in geometric mechanics and geometric optics.