mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Cartan–Kähler theorem is a major result on the

integrability conditions for differential systems In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...

, in the case of

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...

s, for

differential ideal In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exter ...

I

. It is named for

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...

and

Erich Kähler Erich Kähler (; 16 January 1906 – 31 May 2000) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory. Education an ...

Meaning

It is not true that merely having

dI

contained in

I

is sufficient for integrability. There is a problem caused by

singular solution A singular solution ''ys''(''x'') of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the so ...

s. 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 \subseteq M

is a connected, ''

k

''-dimensional, real analytic, regular

integral manifold In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of ...

of ''

I

'' with

r(P) \geq 0

(i.e., the tangent spaces

T_p P

are "extendable" to higher dimensional integral elements). Moreover, assume there is a real analytic submanifold

R \subseteq M

of codimension

r(P)

containing

P

and such that

T_pR \cap H(T_pP)

has dimension

k+1

for all

p \in P

. Then there exists a (locally) unique connected,

(k+1)

-dimensional, real analytic integral manifold

X \subseteq M

I

that satisfies

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 Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...

, ''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

* *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
{{DEFAULTSORT:Cartan-Kahler theorem Partial differential equations Theorems in analysis