In differential geometry, a discipline within mathematics, a distribution on a manifold
In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle
Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds.
These need not be linearly independent at every point, and so aren't formally a vector space basis at every point; thus, the term local generating set can be more appropriate.
Involutive distributions are a fundamental ingredient in the study of integrable systems.
A related idea occurs in Hamiltonian mechanics: two functions
on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.
are thus disjoint, maximal, connected integral manifolds, also called leaves; that is,
In other words, every point admits a foliation chart, i.e. the distribution
Moreover, this local characterisation coincides with the definition of integrability for a
is the group of real invertible upper-triangular block matrices (with
The converse is less trivial but holds by Frobenius theorem.
denotes the set of vector fields spanned by the
[1] Note that, in such case, the associated Lie flag stabilises at a certain point
Any weakly regular distribution has an associated graded vector bundle
Moreover, the Lie bracket of vector fields descends, for any
into a bundle of nilpotent Lie algebras; for this reason,
, however, is in general not locally trivial, since the Lie algebras
[clarification needed] Note that the names (strongly, weakly) regular used here are completely unrelated with the notion of regularity discussed above (which is always assumed), i.e. the dimension of the spaces
is called bracket-generating (or non-holonomic, or it is said to satisfy the Hörmander condition) if taking a finite number of Lie brackets of elements in
is enough to generate the entire space of vector fields on
Clearly, the associated Lie flag of a bracket-generating distribution stabilises at the point
Even though being weakly regular and being bracket-generating are two independent properties (see the examples below), when a distribution satisfies both of them, the integer
In particular, the number of elements in a local basis spanning
, and those vector fields will no longer be linearly independent everywhere.
However, Frobenius theorem does not hold in this context, and involutivity is in general not sufficient for integrability (counterexamples in low dimensions exist).
After several partial results,[5] the integrability problem for singular distributions was fully solved by a theorem independently proved by Stefan[6][7] and Sussmann.
is integrable if and only if the following two properties hold: Similarly to the regular case, an integrable singular distribution defines a singular foliation, which intuitively consists in a partition of
The definition of singular foliation can be made precise in several equivalent ways.
Actually, in the literature there is a plethora of variations, reformulations and generalisations of the Stefan-Sussman theorem, using different notion of singular foliations according to which applications one has in mind, e.g. Poisson geometry[10][11] or non-commutative geometry.