Given an
exterior differential system In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exter ...
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. ...
and
Masatake Kuranishi.
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 of spheres that have come to be called Lie theory. For instance, the latter subject is L ...
.
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 analysis