# 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 ${\displaystyle I}$.

## History

It is named for Élie Cartan and Erich Kähler.

## Meaning

It is not true that merely having ${\displaystyle dI}$  contained in ${\displaystyle I}$  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 ${\displaystyle (M,I)}$  be a real analytic EDS. Assume that ${\displaystyle P\subseteq M}$  is a connected, ${\displaystyle k}$ -dimensional, real analytic, regular integral manifold of ${\displaystyle I}$  with ${\displaystyle r(P)\geq 0}$  (i.e., the tangent spaces ${\displaystyle T_{p}P}$  are "extendable" to higher dimensional integral elements).

Moreover, assume there is a real analytic submanifold ${\displaystyle R\subseteq M}$  of codimension ${\displaystyle r(P)}$  containing ${\displaystyle P}$  and such that ${\displaystyle T_{p}R\cap H(T_{p}P)}$  has dimension ${\displaystyle k+1}$  for all ${\displaystyle p\in P}$ .

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

## Proof and assumptions

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

## References

• Jean Dieudonné, Eléments d'analyse, vol. 4, (1977) Chapt. XVIII.13
• R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, Springer Verlag, New York, 1991.