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 <math>I</math>. It is named for Élie Cartan and Erich Kähler.

Meaning

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

Moreover, assume there is a real analytic submanifold <math>R \subseteq M</math> of codimension <math>r(P)</math> containing <math>P</math> and such that <math>T_pR \cap H(T_pP)</math> has dimension <math>k+1</math> for all <math>p \in P</math>.

Then there exists a (locally) unique connected, <math>(k+1)</math>-dimensional, real analytic integral manifold <math>X \subseteq M</math> of <math>I</math> that satisfies <math>P \subseteq X \subseteq R</math>.

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.
  • 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