Cartan–Kähler theorem

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 ${\displaystyle I}$. 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

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.

External links

• Alekseevskii, D.V. (2001) [1994], "Pfaffian problem", Encyclopedia of Mathematics, EMS Press
• R. Bryant, "Nine Lectures on Exterior Differential Systems", 1999
• E. Cartan, "On the integration of systems of total differential equations," transl. by D. H. Delphenich
• E. Kähler, "Introduction to the theory of systems of differential equations," transl. by D. H. Delphenich