Cartan–Kähler theorem

Remarks

The proof builds $N$ one dimension at a time. Start from a point. An integral manifold of dimension $k$, tangent to $E_k$, is extended to one of dimension $k + 1$, tangent to $E_{k+1}$. This step solves an initial value problem with the Cauchy–Kovalevskaya theorem. Regularity of $E_k$ ensures that the equations to solve are in Cauchy–Kovalevskaya form, and closure of $\mathcal{I}$ under $d$ ensures that the new manifold is again integral. At each step there is freedom in the choice of initial data, and counting it gives the size of the space of solutions.

Example: let $\mathcal{I}$ be generated by $1$-forms $\theta^1, \dots, \theta^s$, independent at each point of an $m$-dimensional manifold. If $d\theta^a \equiv 0$ modulo $\theta^1, \dots, \theta^s$, then every $(m - s)$-plane in the kernel of the $\theta^a$ is ordinary. The theorem then gives integral manifolds of dimension $m - s$, as does the Frobenius theorem, which holds without analyticity.

Analyticity is needed in general. The method rests on the Cauchy–Kovalevskaya theorem, which has no smooth version. Lewy's example gives a linear partial differential equation with polynomial coefficients and a smooth right-hand side with no solution near any point. Writing such an equation as an exterior differential system shows that the smooth analogue of the theorem fails.

Going deeper

For a student

The theorem is the main existence result of the theory of exterior differential systems. It contains the Cauchy–Kovalevskaya theorem and, in the analytic case, the Frobenius theorem. It was used, for instance, to show that every real-analytic Riemannian manifold of dimension $n$ embeds locally isometrically into Euclidean space of dimension $n(n+1)/2$, the Cartan–Janet theorem. In practice one first checks Cartan's test, prolonging the system if the test fails, and then applies the theorem.