Distribution

Definition

Let $M$ be a connected differential manifold of dimension $n$. A $k$-dimensional distribution over $M$ is defined in the following equivalent ways:

• It is a section of the Grassmannian bundle of type $k$
• It is a smooth association of, to each point in $M$, a $k$-dimensional subspace of the tangent space at the point. By smooth here we mean that for every point, there is a neighbourhood where it can be generated by $k$ vector fields.