There seems to be a lot of confusion (among my colleagues, and also of myself) concerning the relationships between singular foliations and integrable p-vector fields (a.k.a. Nambu structures). The aim of this note is to make some clarifications.

**1) How to construct a singular foliation from an integrable p-vector field ?**

The obvious (but stupid) way is to make only 2 kinds of leaves: regular leaves (where the p-vector field does not vanish) and 0-dimensional leaves (where it vanishes). This is stupid, because general singular foliations will have singular leaves of positive dimension, and not just zero-rank points.

A smarter way is to define the rank of a point with respect to an integrable p-vector field Lambda as follows

rank_Lambda (x) = max d such that there is a local coordinate system (y_i) and a (p-d)-vector field Pi which is invariant w.r.t. \partial y_1, …, \partial y_d, such that:

**Proposition.** *Given Lambda, there is a unique singular foliation F such that Lambda is tangent to F, and the rank of every point wrt F is the same as its rank wrt to Lambda.*

We will say that F is generated by Lambda. Of course, F can have leaves of any dimension from 0 to the rank of Lambda.

**2) How to create an integrable p-vector field from a singular foliation ?**

The following process works well locally, at least for analytic foliations (generated by analytic vector fields):

Take an arbitrary family of p vector fields X_1, …, X_p which are tangent to the foliation, and which are linearly independent almost everywhere, then put

The problem of Pi is that it may be non-reduced, in the sense that it vanishes at too many points (or has multiplicity at the points where it vanishes). But it’s not a big problem, because we can divide Pi by a function to make it reduced. For example, if the zero set of Pi is of codimension 1, while the singular set of F has codimension at least 2, then there is a function f which vanishes on the zero set of Pi (without multiplicities), and we can write Pi = f Lambda, where Lambda is still tangent to F.

Some precise definitions:

Pi is called tangent to F is Pi vanishes at every singular point of F, and P gives the tangent space to F at the points where it does not vanish (Pi is allowed to vanish at some regular points of F too)

Pi is called associated to F if Pi is tangent to F, and for any other Lambda tangent to F we have Lambda = g Pi where g is a regular function

**Proposition.** Locally, up to multiplication by a function which is non-zero everywhere, there exists a unique associated integrable p-vector field

**Corollary**. the sheaf of local tangent integrable p-vector field to a foliation F is a locally free of rank 1, i.e. is a line bundle.

Of course, the existence of a global associated integrable p-vector field is equivalent to the global triviality of the above line bundle.

**3) Saturation of foliations
**

Two foliations are called **almost the same** if their tangent spaces coincide almost everywhere.

The composed map

folitation F_1 -> associated integrable p-vector field -> generated foliation F_2

is not an identity map, but is almost identity: F_2 is almost the same as F_1.

The process F_1 -> F_2 is a kind of saturation

**Proposition ?** Each leaf of F_1 lies in a leaf of F_2

**Proposition ?** If repeats the process F_1 -> F_2-> F_3 then in fact F_3 = F_2

**Division lemma ?** If Lambda is integrable and \partial x_1 \wedge \Lambda = 0 then locally \Lambda = \partial x_1 \wedge \Pi where \Pi is invariant wrt \partial x_1.

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