Frobenius theorem (differential topology)

In mathematics , Frobenius' theorem gives an easily checked, equivalent condition for the complete integrability of hyperplane fields , i.e. for the existence of a maximum set of independent solutions to an underdetermined system of partial differential equations .

It was proven by Ferdinand Georg Frobenius in 1877 . In it he deals with Pfaff's problem in the event that the Jacobi determinant of the system and some subsystems vanishes.

Complete integrability

A sub-vector bundle

${\displaystyle F\subset TM}$

of the tangential bundle of a differentiable manifold is called completely integrable (often only integrable ) if there is a foliation of with ${\displaystyle {\mathcal {F}}}$ ${\displaystyle M}$

${\displaystyle F=T{\mathcal {F}}}$

gives.

Frobenius' theorem

Let be a differentiable manifold. The Frobenius theorem states that a sub-vector bundle is fully integrated if and only if the vector fields with values in a Lie subalgebra of the Lie algebra form of all vector fields, ie when the commutator of two valent vector fields again values has. ${\displaystyle M}$ ${\displaystyle F\subset TM}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle F}$

The theorem applies unchanged under the assumption that there is an (infinitely dimensional) Banach manifold . ${\displaystyle M}$

Formulation using differential forms

Let the ring of differential forms on . Consider the ideal for the sub-vector bundle ${\displaystyle \Omega (M)}$ ${\displaystyle M}$ ${\displaystyle F\subset TM}$

${\displaystyle I(F):=\left\{\alpha \in \Omega (M)\mid \forall \ v\in F\colon \iota _{v}\alpha =0\right\}=\left\{\alpha \in \Omega (M)\mid \forall \ v\in F\colon \alpha (v,.,\dotsc ,.)\cong 0\right\}}$.

Then Frobenius' theorem is equivalent to the following statement:

${\displaystyle F\subset TM}$is completely integrable if and only if is closed under the outer derivative , i.e. if it always follows from. ${\displaystyle I(F)}$ ${\displaystyle \alpha \in I(F)}$ ${\displaystyle {\text{d}}\alpha \in I(F)}$

Local description

In local coordinates on an open subset of a hyperplane field of codimension can by 1-forms describing the produce. The hyperplane field can then be integrated if and only if it has 1-forms with ${\displaystyle U\subset M}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle \omega _{1},\dotsc ,\omega _{k}}$ ${\displaystyle I(F)}$ ${\displaystyle U}$ ${\displaystyle \eta _{ij}}$

${\displaystyle {\text{d}}\omega _{i}=\Sigma _{j}\eta _{ij}\wedge \omega _{j}}$

gives.

This in turn is with

${\displaystyle \Omega :=\omega _{1}\wedge \ldots \wedge \omega _{k}}$

equivalent to any of the following conditions:

• for true ${\displaystyle i=1,\dotsc ,k}$

${\displaystyle d\omega _{i}\wedge \Omega =0}$.

• There is a 1-form with ${\displaystyle \alpha }$

${\displaystyle {\text{d}}\Omega =\alpha \wedge \Omega }$.

• There are locally defined functions with ${\displaystyle f_{ij},g_{j}\;(i,j=1,\dotsc ,k)}$

${\displaystyle \omega _{i}=\Sigma _{j}f_{ij}dg_{j}}$.

example

If there is a 1-dimensional hyperplane field (i.e. a straight line field), then all commutators -valent vector fields are zero, so the requirement of Frobenius' theorem is fulfilled trivially. It is found that every straight line field can be integrated. But this already follows directly from the existence and uniqueness theorem for ordinary differential equations , which is also used in the proof of Frobenius' theorem. ${\displaystyle F}$ ${\displaystyle F}$

literature

• Shlomo Sternberg: Lectures on differential geometry. Second edition. With an appendix by Sternberg and Victor W. Guillemin. Chelsea Publishing Co., New York 1983. ISBN 0-8284-0316-3 .

Web links

• Yum Tong Siu: Partial differential equations with compatibility conditions. Harvard Seminar, 2014, PDF (Access Denied).

Individual evidence

1. ↑ Frobenius: About the Pfaff problem. Journal for pure and applied mathematics, Volume 82, 1877, pp. 230-315, digitized.
2. ^ R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 , p. 326 ff.