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 $ | ||
| Line 28: | Line 25: | ||
with $ p $ | with $ p $ | ||
$ C ^ {r} $ | $ C ^ {r} $ | ||
| − | vector fields $ X _ {1} \dots X _ {p} $ | + | vector fields $ X _ {1}, \dots, X _ {p} $ |
| − | on it for which the vectors $ X _ {1} ( y) \dots X _ {p} ( y) $ | + | on it for which the vectors $ X _ {1} ( y), \dots, X _ {p} ( y) $ |
form a basis of the space $ D ( y) $ | form a basis of the space $ D ( y) $ | ||
at each point $ y \in U $. | at each point $ y \in U $. | ||
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$$ | $$ | ||
| − | where $ \omega ^ {p+} | + | 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 | ||
Latest revision as of 10:11, 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) |
Involutive distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involutive_distribution&oldid=47431