Difference between revisions of "Cartan lemma"
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| + | $#C+1 = 9 : ~/encyclopedia/old_files/data/C020/C.0200520 Cartan lemma | ||
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| − | + | If for $ 2p $ | |
| + | linear forms $ \phi _ {i} , \sigma ^ {i} $, | ||
| + | $ i = 1 \dots n $, | ||
| + | in $ n $ | ||
| + | variables the sum of the exterior products vanishes | ||
| − | + | $$ | |
| + | \sum _ {i = 1 } ^ { p } | ||
| + | \phi _ {i} \wedge \sigma ^ {i} = 0, | ||
| + | $$ | ||
| + | |||
| + | and if the $ \sigma ^ {i} $ | ||
| + | are linearly independent, then the $ \phi _ {i} $ | ||
| + | are linear combinations of the $ \sigma ^ {i} $ | ||
| + | with symmetric coefficients: | ||
| + | |||
| + | $$ | ||
| + | \phi _ {i} = \sum | ||
| + | a _ {ij} \sigma ^ {j} ,\ \ | ||
| + | a _ {ij} = a _ {ji} . | ||
| + | $$ | ||
Proved by E. Cartan in 1899. | Proved by E. Cartan in 1899. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
Latest revision as of 10:08, 4 June 2020
If for $ 2p $
linear forms $ \phi _ {i} , \sigma ^ {i} $,
$ i = 1 \dots n $,
in $ n $
variables the sum of the exterior products vanishes
$$ \sum _ {i = 1 } ^ { p } \phi _ {i} \wedge \sigma ^ {i} = 0, $$
and if the $ \sigma ^ {i} $ are linearly independent, then the $ \phi _ {i} $ are linear combinations of the $ \sigma ^ {i} $ with symmetric coefficients:
$$ \phi _ {i} = \sum a _ {ij} \sigma ^ {j} ,\ \ a _ {ij} = a _ {ji} . $$
Proved by E. Cartan in 1899.
References
| [1] | E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945) |
Comments
The original paper containing this result is [a1].
References
| [a1] | E. Cartan, "Sur certaines expressions différentielles et le problème de Pfaff" Ann. Ec. Norm. (3) , 16 (1899) pp. 239–332 |
How to Cite This Entry:
Cartan lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_lemma&oldid=46261
Cartan lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_lemma&oldid=46261
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article