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 I. It is named for Élie Cartan and Erich Kähler.

Meaning

It is not true that merely having dI contained in I 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.

Statement

Let (M,I) be a real analytic EDS. Assume that is a connected, k-dimensional, real analytic, regular integral manifold of I with r(P) \geq 0 (i.e., the tangent spaces T_p P are "extendable" to higher dimensional integral elements). Moreover, assume there is a real analytic submanifold of codimension r(P) containing P and such that has dimension k+1 for all p \in P. Then there exists a (locally) unique connected, (k+1)-dimensional, real analytic integral manifold of I that satisfies.

Proof and assumptions

The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.