In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals
It is named for Élie Cartan and Erich Kähler.
contained in
is sufficient for integrability.
There is a problem caused by singular solutions.
The theorem computes certain constants that must satisfy an inequality in order that there be a solution.
be a real analytic EDS.
-dimensional, real analytic, regular integral manifold of
(i.e., the tangent spaces
are "extendable" to higher dimensional integral elements).
Moreover, assume there is a real analytic submanifold
Then there exists a (locally) unique connected,
-dimensional, real analytic integral manifold
The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.