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Let $M$ be an $n$-dimensional manifold. A distribution $D$ on $M$ is an assignment of a subspace $D_m \subset T_mM$ for each $m\in M$. A vector field $X$ on M belongs to $D$ if $X_m \in D_m$, for all $m\in M$. A distribution $D$ is said to be involutive if $[X,Y]\in D$ for all $X,Y\in D$. A distribution $D$ is called non-singular if for all $m\in M$, there exist an open neighborhood $U$ of $m$ such that the dimension of $D$ is constant on $U$.

Under which conditions is a non-singular distribution involutive?

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    $\begingroup$ These are independent conditions. Checking involutivity involves looking at derivatives of vector fields that span the distribution, while non-singularity does not involve any information about the derivatives. So in any case, one would not expect a simple relationship between them. $\endgroup$
    – Igor Khavkine
    Commented Nov 2, 2017 at 8:18
  • $\begingroup$ Obviously the Frobenius theorem holds for sufficiently smooth distributions: involutivity just when the tangent bundle of a foliation. I don't know of any result on involutivity which holds for rough distributions. I doubt if you can prove anything without some Sobolev or Hoelder condition on $D$. $\endgroup$
    – Ben McKay
    Commented Nov 2, 2017 at 10:37
  • $\begingroup$ Your definition should state some smoothness condtion on $X$, since otherwise $[X,Y]$ might be nowhere defined. $\endgroup$
    – Ben McKay
    Commented Nov 2, 2017 at 10:38

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Obviously the Frobenius theorem holds for sufficiently smooth distributions: involutivity just when the tangent bundle of a foliation. The most recent results on generalizations of the Frobenius theorem to nonsmooth distributions appear, along with an extensive bibliography of a raft of similar generalizations, in:

MR3530282 Reviewed Luzzatto, S.(I-ICTP); Türeli, S.(I-ICTP); War, K. M.(I-ICTP) A Frobenius theorem for corank-1 continuous distributions in dimensions two and three. (English summary) Internat. J. Math. 27 (2016), no. 8, 1650061, 30 pp. 58A30

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