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



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> 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 $T\_pM$.
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(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.



Rank of a distribution
(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 $\pi^\*\beta = \alpha$.


