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Difference between revisions of "Involutive distribution"

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The geometric interpretation of a completely-integrable differential system on an  $  n $-
+
The geometric interpretation of a completely-integrable differential system on an  $  n $-dimensional differentiable manifold  $  M  ^ {n} $
dimensional differentiable manifold  $  M  ^ {n} $
 
 
of class  $  C  ^ {k} $,  
 
of class  $  C  ^ {k} $,  
 
$  k \geq  3 $.  
 
$  k \geq  3 $.  
A  $  p $-
+
A  $  p $-dimensional distribution (or a differential system of dimension  $  p $)  
dimensional distribution (or a differential system of dimension  $  p $)  
 
 
of class  $  C  ^ {r} $,  
 
of class  $  C  ^ {r} $,  
 
$  1 \leq  r < k $,  
 
$  1 \leq  r < k $,  
 
on  $  M  ^ {n} $
 
on  $  M  ^ {n} $
 
is a function associating to each point  $  x \in M  ^ {n} $
 
is a function associating to each point  $  x \in M  ^ {n} $
a  $  p $-
+
a  $  p $-dimensional linear subspace  $  D( x) $
dimensional linear subspace  $  D( x) $
 
 
of the tangent space  $  T _ {x} ( M  ^ {n} ) $
 
of the tangent space  $  T _ {x} ( M  ^ {n} ) $
 
such that  $  x $
 
such that  $  x $
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$$
 
$$
  
where  $  \omega  ^ {p+} 1 \dots \omega  ^ {n} $
+
where  $  \omega  ^ {p+1} \dots \omega  ^ {n} $
are  $  1 $-
+
are  $  1 $-forms of class  $  C  ^ {r} $,  
forms of class  $  C  ^ {r} $,  
 
 
linearly independent at each point  $  x \in U $;  
 
linearly independent at each point  $  x \in U $;  
 
in other words,  $  D $
 
in other words,  $  D $
 
is locally equivalent to the system of differential equations  $  \omega  ^  \alpha  = 0 $.  
 
is locally equivalent to the system of differential equations  $  \omega  ^  \alpha  = 0 $.  
 
Then  $  D $
 
Then  $  D $
is an involutive distribution if there exist  $  1 $-
+
is an involutive distribution if there exist  $  1 $-forms  $  \omega _  \beta  ^  \alpha  $
forms  $  \omega _  \beta  ^  \alpha  $
 
 
on  $  U $
 
on  $  U $
 
such that
 
such that

Revision as of 10:09, 21 March 2022


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 (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)
How to Cite This Entry:
Involutive distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involutive_distribution&oldid=47431
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article