What is a differentiable distribution on a manifold

Question

Assume $M$ is a smooth manifold and $d:u\to Q_u<T_uM$ is a distribution on $M$.

What does "$d$ is a differentiable distribution" mean? What does it mean for $Q_u$ to depend smoothly on $u$?

Answer 1

As you wrote a distribution is a choice of a subspace $Q_u$ in each tangent space $T_uM$ such that $\dim Q_u$ is constant, say $k$. The distribution is smooth if around each point $u_0$ there are $k$ smooth vector fields $X_1,\cdots,X_k$ such that $Q_u = \rm span(X_1(u),\cdots,X_k(u)) \rm $ . Namely, the distribution is locally generated by a smooth basis.