Frobenius’ theorem

Theorem (Frobenius).

Let M be a smooth manifold (C) and let Δ be a distributionDlmfPlanetmath on M. Then Δ is completely integrable if and only if Δ is involutive.

One direction in the proof is pretty easy since the tangent space of an integral manifold is involutive, so sometimes the theorem is only stated in one direction, that is: If Δ is involutive then it is completely integrable.

Another way to the theorem is that if we have n vector fields {Xk}k=1n on a manifold M such that they are linearly independentMathworldPlanetmath at every point of the manifold, and furthermore if for any k,m we have [Xk,Xm]=j=1najXj for some C functionsMathworldPlanetmath aj, then for any point xN, there exists a germ of a submanifold NM, through x, such that TN is spanned by {Xk}k=1n. Note that if we extend N to all of M, it need not be an embedded submanifold anymore, but just an immersed one.


  • 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
  • 2 Frobenius theoremMathworldPlanetmath at Wikipedia:
Title Frobenius’ theorem
Canonical name FrobeniusTheorem
Date of creation 2013-03-22 14:52:03
Last modified on 2013-03-22 14:52:03
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 9
Author jirka (4157)
Entry type Theorem
Classification msc 37C10
Classification msc 53-00
Classification msc 53B25
Related topic Distribution5
Related topic IntegralManifold