Cartan–Kähler theorem
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]\displaystyle{ I }[/math]. It is named for Élie Cartan and Erich Kähler.
Meaning
It is not true that merely having [math]\displaystyle{ dI }[/math] contained in [math]\displaystyle{ 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]\displaystyle{ (M,I) }[/math] be a real analytic EDS. Assume that [math]\displaystyle{ P \subseteq M }[/math] is a connected, [math]\displaystyle{ k }[/math]-dimensional, real analytic, regular integral manifold of [math]\displaystyle{ I }[/math] with [math]\displaystyle{ r(P) \geq 0 }[/math] (i.e., the tangent spaces [math]\displaystyle{ T_p P }[/math] are "extendable" to higher dimensional integral elements).
Moreover, assume there is a real analytic submanifold [math]\displaystyle{ R \subseteq M }[/math] of codimension [math]\displaystyle{ r(P) }[/math] containing [math]\displaystyle{ P }[/math] and such that [math]\displaystyle{ T_pR \cap H(T_pP) }[/math] has dimension [math]\displaystyle{ k+1 }[/math] for all [math]\displaystyle{ p \in P }[/math].
Then there exists a (locally) unique connected, [math]\displaystyle{ (k+1) }[/math]-dimensional, real analytic integral manifold [math]\displaystyle{ X \subseteq M }[/math] of [math]\displaystyle{ I }[/math] that satisfies [math]\displaystyle{ 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.
External links
- Hazewinkel, Michiel, ed. (2001), "Pfaffian problem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p072530
- 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
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