# Cartan–Kähler theorem

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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 $I$ .

## History

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 of the theorem

Let $(M,I)$  be a real analytic EDS. Assume that $P\subseteq M$  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 $R\subseteq M$  of codimension $r(P)$  containing $P$  and such that $T_{p}R\cap H(T_{p}P)$  has dimension $k+1$  for all $p\in P$ .

Then there exists a (locally) unique connected, $(k+1)$ -dimensional, real analytic integral manifold $X\subseteq M$  of $I$  that satisfies $P\subseteq X\subseteq R$ .

## Proof and assumptions

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