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 .

HistoryEdit

It is named for Élie Cartan and Erich Kähler.

MeaningEdit

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 of the theoremEdit

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 assumptionsEdit

The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.

ReferencesEdit

  • 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 linksEdit

  • Alekseevskii, D.V. (2001) [1994], "Pfaffian problem", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • 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