The Frobenius integrability theorem states that a smooth constant-rank distribution on a manifold is tangent to a local foliation
by integral manifolds precisely when it is an
involutive distribution. Equivalently,
if two local vector fields 
and 
are tangent to the distribution, then their Lie
bracket ![[X,Y]](/images/equations/FrobeniusIntegrabilityTheorem/Inline3.svg)
must also be tangent to it. The theorem therefore turns
this differential condition into the existence of local solution manifolds.
Frobenius Integrability Theorem
See also
Cartan Distribution, Foliation, Integral Manifold, Involutive Distribution, Lie Bracket, Vector FieldExplore with Wolfram|Alpha
References
Lee, J. M. Introduction to Smooth Manifolds, 2nd ed. New York: Springer, 2012.Cite this as:
Weisstein, Eric W. "Frobenius Integrability Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FrobeniusIntegrabilityTheorem.html