Frobenius theorem (differential topology)

In mathematics, the Frobenius theorem is an easy nachzuprüfende, equivalent condition for the complete integrability of hyperplane fields, ie for the existence of a maximum set of independent solutions to a certain system of partial differential equations.

Complete integrability

A sub-vector bundle

Of the tangent bundle of a differentiable manifold is called completely integrable (often just integrable ) if there is a foliation of with

Frobenius theorem

The Frobenius theorem states that a sub-vector bundle if and only completely integrable if the vector fields form with values ​​in a Lie subalgebra of the Lie algebra of all vector fields, ie, if the commutator of two -valued vector fields again has values ​​.

Formulation using differential forms

Be the ring of differential forms. For sub-vector bundle, consider the ideal

Then the Frobenius theorem equivalent to the following statement:

If and only completely integrable if is closed under the exterior derivative, so if follows from always.

Local Description

In local coordinates on an open subset can be a hyperplane field of codimension by 1- forms describe which generate. The hyperplane field is then exactly integrable if it with 1- forms

There.

This in turn is connected

Equivalent to any of the following conditions:

• Applies for

• There is a 1 - shape with

• There are locally defined functions with

Example

If a 1-dimensional hyperplane field (ie, a straight line box ), then the hypothesis of the theorem of Frobenius are all commutators -valued vector fields zero, so trivially satisfied. This gives that every straight field is integrable. However, this already follows directly from the existence and uniqueness theorem for ordinary differential equations, which is also used in the proof of the theorem of Frobenius.