A rank- distribution on a smooth manifold is a rank- vector subbundle of its tangent bundle. It is involutive when the Lie bracket of vector fields of any two local sections is again a section.
A distribution is integrable when every point lies on an immersed submanifold whose tangent spaces equal the distribution. Such submanifolds are its integral manifolds.
If has constant rank, the Frobenius theorem says that is integrable exactly when its annihilator is closed under the exterior derivative: equivalently,This is also equivalent to involutivity of .
For a nowhere-zero 1-form , the hyperplane distribution is tangent to hypersurfaces exactly when . In general relativity this characterizes hypersurface-orthogonal vector fields and implies vanishing twist.
A plane distribution on a three-manifold is a rank-two distribution. Locally it is the kernel of a nowhere-zero differential 1-form .
The Frobenius theorem gives the local criterion
A contact structure on a three-manifold is a plane distribution locally of the form with nowhere zero. It is maximally nonintegrable; the standard example on is .
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In differential geometry, the term "distribution" refers to a smooth assignment of a subspace of the tangent space at each point of a manifold. More formally, given a smooth manifold \( M \), a distribution is a smooth assignment of a vector subspace \( D_p \) of the tangent space \( T_p M \) at each point \( p \in M \). Distributions are often used to study geometric structures, such as foliations and control systems.