An involutive distribution is a constant-rank choice of subspace 
of each tangent space 
such that the Lie bracket ![[X,Y]](/images/equations/InvolutiveDistribution/Inline3.svg)
lies in the distribution
whenever the vector fields 
and 
do. In other words, the vector
fields tangent to 
are closed under the Lie bracket.
The Frobenius integrability theorem
states that this condition is equivalent to the local existence of integral
manifolds tangent to 
.
Involutive Distribution
See also
Cartan Distribution, Frobenius Integrability Theorem, Integral Manifold, Lie Bracket, Rank, Tangent Space, 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. "Involutive Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InvolutiveDistribution.html