Cartan–Kuranishi prolongation theorem

Given an exterior differential system defined on a manifold M, the Cartan–Kuranishi prolongation theorem says that after a finite number of prolongations the system is either in involution (admits at least one 'large' integral manifold), or is impossible.

History

The theorem is named after Élie Cartan and Masatake Kuranishi. Cartan made several attempts in 1946 to prove the result, but it was in 1957 that Kuranishi provided a proof of Cartan's conjecture.[1]

Applications

This theorem is used in infinite-dimensional Lie theory.

See also

• Cartan-Kähler theorem

References

1. Bryant, Robert L.; Chern, S. S.; Gardner, Robert B.; Goldschmidt, Hubert L.; Griffiths, P. A. (2013-06-29). Exterior Differential Systems. Springer Science & Business Media. ISBN 978-1-4613-9714-4.

• M. Kuranishi, On É. Cartan's prolongation theorem of exterior differential systems, Amer. J. Math., vol. 79, 1957, p. 1–47
• "Partial differential equations on a manifold", Encyclopedia of Mathematics, EMS Press, 2001 [1994]