Cartan lemma

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