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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.