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 . 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[edit]

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.

External links[edit]

  • 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