Distribution and Frobenius Theorem
Distribution
Given a smooth manifold
, a distribution of -planes on is a subbundle of the tangent bundle , such that for each , the fiber is a -dimensional linear subspace of .
Integrable Distribution
A distribution
of -planes is integrable if for all , there exists a nonempty immersed submanifold in of dimension containing such that .
Frobenius Theorem
A distribution
of -planes on is integrable if and only if the space of vector fields along , i.e. is a (sub) Lie algebra closed under the Lie bracket operation. In other words, is involutive.
Corollary
Suppose
is a smooth manifold, is an involutive -plane distribution on , and is a codimension- embedded submanifold. If is a point such that is complementary to , then there is a flat chart for centered at in which is the slice .
Alternative Form of Frobenius Theorem
Let
be a -dimensional, involutive distribution on a smooth -manifold . Let . Then there exists an integral manifold of passing through . Indeed, there exists a cubic coordinate system which is centered at , with coordinate functions such that the slices are integral manifolds of ; and if is a connected integral manifold of such that , then lies in one of these slices.
Foliations
Foliation
Let
be an -dimensional manifold. A -dimensional foliation is a decomposition of as a disjoint union of connected immersed submanifolds of : where each is called a leaf, and the following condition is satisfied: Every point has a neighbourhood and a local chart such that for each leaf , is described by the level sets of . That is, , where are real constants. 