1. Distributions on manifolds

Let M be a smooth manifold. Assume that for each p ∈ M a subspace Δ(p) ⊂ T_{p}M is assigned such that 1) dim Δ(p) = k is constant, 2) for each q ∈ M there exists a neigbourhood U with vector fields E_{1}, E_{2}, …, E_{k} ∈ 𝔛(U) such that, for each q ∈ U, Δ(q) = span{E_{1}(q), E_{2}(q), …, E_{k}(q)}. We say that the dimension of the distribution Δ equals k.

A distribution Δ on a manifold M is called totally integrable if for each p ∈ M there exists a submanifold Σ passing through p such that T_{q}Σ = Δ(q), for any q ∈ Σ. This submanifold is called an integral submanifold of Δ. A totally integrable distibution is called a foliation, and the integral submanifolds are called the leaves.

Locally, a distribution Δ can be given either by vector fields E_{a}, a = 1, …, k, spanning the subspaces Δ(p) ⊂ T_{p}M, or by differential forms ω^{1}, ω^{2}, …, ω^{n−k} such that, for any V ∈ T_{p}M, V ∈ Δ(p) if and only if ω^{α}(V) = 0, α = 1, …, n − k

The following theorem gives necessary and sufficient conditions for a distribution be completely integrable.

Theorem 1 (Frobenius). Let Δ be a distribution on a manifold M locally given by a frame field E_{a}, a = 1, …, k, or by the fields of 1-forms ω^{α}. The following conditions are equivalent:

a) The distribution Δ is completely integrable;

b) [E_{a}, E_{b}] = Q_{ab}^{c}E_{c}, where Q_{ab}^{c} are functions;

c) dω^{γ} = Q_{β}^{γ} ∧ ω^{β}, where Q_{β}^{γ} are 1-forms.

Example 1.1. Let a distribution Δ on ℝ^{3} be given by the 1-form ω = dz − xdx − ydy. Then dω = 0, and Δ is completely integrable because this distribution consists of panes tangent to hyperbolic paraboloids of the family z = \frac{1}{2}(x^{2} + y^{2}).

Example 1.2. Let a distribution Δ on ℝ^{3} be given by the 1-form ω = dy − zdx. Then one can take the local frame spanning Δ as follows: E_{1} = \frac{∂}{∂x} + z \frac{∂}{∂y}, E_{2} = \frac{∂}{∂z}. Then [E_{1}, E_{2}] = −\frac{∂}{∂y}, and this vector field does not lie in Δ, because ω(\frac{∂}{∂y}) = −1 ≠ 0. Thus this distribution is not completely integrable. This means that one cannot find a surface tangent to Δ.

2. Non-holonomic surface in E^{3}

Let E_{3} be a three-dimensional Euclidean space, and Δ be a distribution on E_{3}. We denote by (·, ·) the scalar product in E_{3}. This distribution can be given by a unit vector field n⃗ normal to Δ. The integrability condition for Δ can be formulated in terms of n⃗.

Theorem 2. Δ is integrable if and only if (n⃗, rot n⃗) = 0