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I am reading about distributions in the context of differential geometry.

A distribution $S$ of dimension $r$ on a manifold $M$ is an assignment to each point $p \in M$ of an $r$-dimensional subspace $S_p$ of $T_pM$.

(1) Is this $r$ called rank of the distribution?

Further, I introduce a closed $3$-form $\alpha$ on $M$, and consider the distribution $\ker \alpha := \{X \in TM: \alpha(X,{}\cdot{},{}\cdot{}) = 0\}$. I read that if $\ker \alpha$ is of constant rank and has closed leaves, then I can consider the quotient $M/\ker \alpha$ with a structure induced by $\alpha$.

(2) I cannot really understand the last sentence above. Is the condition 'constant rank' necessary to construct the quotient $M/\ker \alpha$? What does 'closed leaves' mean?

Any help about (1) or (2) is appreciated. Thanks.

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    $\begingroup$ The leaves of a distribution $S$ on a manifold $M$ are the maximal dimensional connected immersed submanifolds $L \subset M$ whose tangent spaces $T_p L$ at any point $p \in L$ are the subspaces $T_p L = S_p$. Leaves might not exist, but if there is a leaf of $S$ through a point $p$, then there is a unique leaf of $S$ through $p$. $\endgroup$
    – Ben McKay
    Commented May 25, 2018 at 10:38
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    $\begingroup$ This stuff is covered well in a lot of textbooks on differential geometry, and in particular in Spivak. Perhaps you should read up a little from the textbooks before trying MathOverFlow. $\endgroup$
    – Ben McKay
    Commented May 25, 2018 at 10:39
  • $\begingroup$ @BenMcKay Thanks for the reference and the explanation. $\endgroup$
    – Gibbs
    Commented May 25, 2018 at 13:20
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    $\begingroup$ Here it is on MSE: math.stackexchange.com/questions/2795410/rank-of-a-distribution . First, just because a question gets no attention on MSE, that does not mean that it is suitable for this site. In this case, I agree with the above commentators that this question is not of research level. $\endgroup$
    – j.c.
    Commented May 25, 2018 at 14:00
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    $\begingroup$ Second, if you cross-post a question you should wait at least a few days, and it is good etiquette to link to the previous question so that there is no duplication of effort. You only waited 1 hour before you reposted it here, and you got a good answer on MSE not too long afterwards, so "did not get too much attention" is not a fair assessment. $\endgroup$
    – j.c.
    Commented May 25, 2018 at 14:00

1 Answer 1

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(1) The boxed sentence would be better written as "A distribution $S$ of rank $r$ on a manifold $M$ is an assignment of an $r$-dimensional subspace $S_p$ of $T_pM$ to each point $p\in M$." Confusing 'dimension' and 'rank' in this context is careless writing since, assuming that $S\subset TM$ is a smooth subbundle, it already has a dimension (as a manifold), which would be $s = r + \dim M$.

(2) The problem is that, even when $\ker\alpha\subset TM$ is a (smooth) distribution of rank $r$, its set of leaves, say, $Q$, may not support a structure of a smooth, Hausdorff manifold such that the canonical projection $\pi:M\to Q$ is a smooth submersion. You need more than that the leaves be closed submanifolds of $M$ in order for this to hold; it was careless of the writer of your source only to require the leaves to be closed. Meanwhile, if the set $Q$ does support the structure of a smooth, Hausdorff manifold such that the canonical projection $\pi:M\to Q$ is a smooth submersion, then there will exist a unique $3$-form $\beta$ on $Q$ such that $\pi^*\beta = \alpha$.

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