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The geometric interpretation of a completely-integrable differential system on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i0525501.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i0525502.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i0525503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i0525504.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i0525506.png" />-dimensional distribution (or a differential system of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i0525508.png" />) of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i0525509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255010.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255011.png" /> is a function associating to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255012.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255013.png" />-dimensional linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255014.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255016.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255017.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255018.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255019.png" /> vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255020.png" /> on it for which the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255021.png" /> form a basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255022.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255023.png" />. The distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255024.png" /> is said to be involutive if for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255025.png" />,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255026.png" /></td> </tr></table>
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{{TEX|done}}
  
This condition can also be stated in terms of differential forms. The distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255027.png" /> is characterized by the fact that
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255028.png" /></td> </tr></table>
+
$$
 +
[ X _ {i} , X _ {j} ] ( y)  \in  D ( y) ,\ \
 +
1 \leq  i , j \leq  p .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255029.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255030.png" />-forms of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255031.png" />, linearly independent at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255032.png" />; in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255033.png" /> is locally equivalent to the system of differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255034.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255035.png" /> is an involutive distribution if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255036.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255038.png" /> such that
+
This condition can also be stated in terms of differential forms. The distribution  $  D $
 +
is characterized by the fact that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255039.png" /></td> </tr></table>
+
$$
 +
D ( y)  = \{ {X \in T _ {y} ( M  ^ {n} ) } : {
 +
\omega  ^  \alpha  ( y) ( X) = 0 } \}
 +
,\  p < \alpha \leq  n ,
 +
$$
  
that is, the exterior differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255040.png" /> belong to the ideal generated by the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255041.png" />.
+
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
  
A distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255042.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052550/i05255044.png" /> is involutive if and only if (as a differential system) it is an [[Integrable system|integrable system]] (Frobenius' theorem).
+
$$
 +
d \omega  ^  \alpha  = \
 +
\sum _ {\beta = p + 1 } ^ { n }
 +
\omega  ^  \beta  \wedge \omega _  \beta  ^  \alpha  ,
 +
$$
 +
 
 +
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====
 
====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>
 
<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>

Revision as of 22:13, 5 June 2020


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