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 I {\displaystyle I} . It is named for Élie Cartan and Erich Kähler.
Meaning It is not true that merely having d I {\displaystyle dI} contained in I {\displaystyle I} 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 ( M , I ) {\displaystyle (M,I)} be a real analytic EDS. Assume that P ⊆ M {\displaystyle P\subseteq M} is a connected, k {\displaystyle k} -dimensional, real analytic, regular integral manifold of I {\displaystyle I} with r ( P ) ≥ 0 {\displaystyle r(P)\geq 0} (i.e., the tangent spaces T p P {\displaystyle T_{p}P} are "extendable" to higher dimensional integral elements). Moreover, assume there is a real analytic submanifold R ⊆ M {\displaystyle R\subseteq M} of codimension r ( P ) {\displaystyle r(P)} containing P {\displaystyle P} and such that T p R ∩ H ( T p P ) {\displaystyle T_{p}R\cap H(T_{p}P)} has dimension k + 1 {\displaystyle k+1} for all p ∈ P {\displaystyle p\in P} . Then there exists a (locally) unique connected, ( k + 1 ) {\displaystyle (k+1)} -dimensional, real analytic integral manifold X ⊆ M {\displaystyle X\subseteq M} of I {\displaystyle I} that satisfies P ⊆ X ⊆ R {\displaystyle P\subseteq X\subseteq R} .
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 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