Cartan–Kähler theorem explained

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

. It is named for Élie Cartan and Erich Kähler.

Meaning

It is not true that merely having

dI

contained in

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)

be a real analytic EDS. Assume that

P\subseteqM

is a connected,

k

-dimensional, real analytic, regular integral manifold of

I

 with

r(P)\geq0

(i.e., the tangent spaces

TpP

are "extendable" to higher dimensional integral elements).

Moreover, assume there is a real analytic submanifold

R\subseteqM

of codimension

r(P)

containing

P

and such that

TpR\capH(TpP)

has dimension

k+1

for all

p\inP

.

Then there exists a (locally) unique connected,

(k+1)

-dimensional, real analytic integral manifold

X\subseteqM

of

I

that satisfies

P\subseteqX\subseteqR

.

Proof and assumptions

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

References

External links

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