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
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...
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.
Applications
This theorem is used in infinite-dimensional
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For instance, ...
.
See also
*
Cartan-Kähler theorem
References
* M. Kuranishi, ''On É. Cartan's prolongation theorem of exterior differential systems'', Amer. J. Math., vol. 79, 1957, p. 1–47
*
{{DEFAULTSORT:Cartan-Kuranishi prolongation theorem
Partial differential equations
Theorems in mathematical analysis