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.
It is not true that merely having 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.
Let be a real analytic EDS. Assume that is a connected, -dimensional, real analytic, regular integral manifold of with (i.e., the tangent spaces are "extendable" to higher dimensional integral elements).
Moreover, assume there is a real analytic submanifold of codimension containing and such that has dimension for all .
Then there exists a (locally) unique connected, -dimensional, real analytic integral manifold of that satisfies .
Proof and assumptions
The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.
- 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.
- Alekseevskii, D.V. (2001) , "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