View source for Involutive distribution

<!--
i0525501.png
$#A+1 = 42 n = 0
$#C+1 = 42 : ~/encyclopedia/old_files/data/I052/I.0502550 Involutive distribution
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The geometric interpretation of a completely-integrable differential system on an  $  n $-dimensional differentiable manifold  $  M  ^ {n} $
of class  $  C  ^ {k} $, 
$  k \geq  3 $. 
A  $  p $-dimensional distribution (or a differential system of dimension  $  p $) 
of class  $  C  ^ {r} $, 
$  1 \leq  r < k $, 
on  $  M  ^ {n} $
is a function associating to each point  $  x \in M  ^ {n} $
a  $  p $-dimensional linear subspace  $  D( x) $
of the tangent space  $  T _ {x} ( M  ^ {n} ) $
such that  $  x $
has a neighbourhood  $  U $
with  $  p $
$  C  ^ {r} $
vector fields  $  X _ {1}, \dots, X _ {p} $
on it for which the vectors  $  X _ {1} ( y), \dots, X _ {p} ( y) $
form a basis of the space  $  D ( y) $
at each point  $  y \in U $. 
The distribution  $  D $
is said to be involutive if for all points  $  y \in U $,

$$ 
[ X _ {i} , X _ {j} ] ( y)  \in  D ( y) ,\ \ 
1 \leq  i , j \leq  p .
$$

This condition can also be stated in terms of differential forms. The distribution  $  D $
is characterized by the fact that

$$ 
D ( y)  =  \{ {X \in T _ {y} ( M  ^ {n} ) } : {
\omega  ^  \alpha  ( y) ( X) = 0 } \}
,\  p < \alpha \leq  n ,
$$

where  $  \omega  ^ {p+1}, \dots, \omega  ^ {n} $
are  $  1 $-forms of class  $  C  ^ {r} $, 
linearly independent at each point  $  x \in U $; 
in other words,  $  D $
is locally equivalent to the system of differential equations  $  \omega  ^  \alpha  = 0 $. 
Then  $  D $
is an involutive distribution if there exist  $  1 $-forms  $  \omega _  \beta   ^  \alpha  $
on  $  U $
such that

$$ 
d \omega  ^  \alpha   = \ 
\sum _ {\beta = p + 1 } ^ { n } 
\omega  ^  \beta  \wedge \omega _  \beta   ^  \alpha  ,
$$

that is, the exterior differentials  $  d \omega  ^  \alpha  $
belong to the ideal generated by the forms  $  \omega  ^  \beta  $.

A distribution  $  D $
of class  $  C  ^ {r} $
on  $  M  ^ {n} $
is involutive if and only if (as a differential system) it is an [[Integrable system|integrable system]] (Frobenius' theorem).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Narasimhan,   "Analysis on real and complex manifolds" , North-Holland &amp; Masson  (1968)  (Translated from French)</TD></TR></table>