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.
Meaning 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.
Statement 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.
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
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