Cartan lemma
From Encyclopedia of Mathematics
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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.
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