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.
1) $r$ is the rank of the distribution.
2) constant rank is not necessary to construct the quotient $M/ker\alpha$. Consider $(x^2+y^2+z^2)dx\wedge dy\wedge dz$ defined on $\mathbb{R}^3$, its does not have a constant rank but its leaves are points, so the quotient space $\mathbb{R}^3/\ker\alpha$ exists.
More generally, regular foliation are studied. In this case this means that the rank of the distribution is constant and it is integrable i.e there exists submanifolds tangent to the distribution called the leaves of the foliation. This follows here from the Frobenius theorem. We also need the fact that the leaves are closed. The typical example here will be torus on which you define a foliation by irrational lines (the quotient space does not exists as a manifold are least) or a foliation by circles for which the quotient space exists.