Involutive distribution
The geometric interpretation of a completely-integrable differential system on an -dimensional differentiable manifold
of class
,
. A
-dimensional distribution (or a differential system of dimension
) of class
,
, on
is a function associating to each point
a
-dimensional linear subspace
of the tangent space
such that
has a neighbourhood
with
vector fields
on it for which the vectors
form a basis of the space
at each point
. The distribution
is said to be involutive if for all points
,
![]() |
This condition can also be stated in terms of differential forms. The distribution is characterized by the fact that
![]() |
where are
-forms of class
, linearly independent at each point
; in other words,
is locally equivalent to the system of differential equations
. Then
is an involutive distribution if there exist
-forms
on
such that
![]() |
that is, the exterior differentials belong to the ideal generated by the forms
.
A distribution of class
on
is involutive if and only if (as a differential system) it is an integrable system (Frobenius' theorem).
References
[1] | C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) |
[2] | R. Narasimhan, "Analysis on real and complex manifolds" , North-Holland & Masson (1968) (Translated from French) |
Involutive distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involutive_distribution&oldid=11229