Cartan-Kähler theorem with proof (Bryant)

Theorem 2.2 (Cartan–Kähler). Let $I \subset Ω^{*} (M)$ be a real analytic differential ideal. Let $P \subset M$ be a connected, $p$ -dimensional, real analytic, Kähler-regular integral manifold of $I$ .
Suppose that $r = r (P)$ is a non-negative integer. Let $R \subset M$ be a real analytic submanifold of $M$ which is of codimension $r$ , which contains $P$ , and which satisfies the condition that $T_{x} R$ and $H (T_{x} P)$ are transverse in $T_{x} M$ for all $x \in P$ .
Then there exists a real analytic integral manifold of $I$ , $X$ , which is connected and $(p + 1)$ -dimensional and which satisfies $P \subset X \subset R$ . This manifold is unique in the sense that any other real analytic integral manifold of $I$ with these properties agrees with $X$ on an open neighborhood of $P$ .

Proof. The theorem is local, so it suffices to prove existence and uniqueness in a neighborhood of a single point $x_{0} \in P$ . Let $s = \dim M - (r + p + 1)$ . (The following proof holds with the obvious simplifications if any of $p$ , $r$ , or $s$ are zero. For simplicity of notation, we assume that they are all positive.)

Our hypothesis implies that the vector space $T_{x} R \cap H (T_{x} P)$ has dimension $p + 1$ for all $x \in P$ . It follows that we may choose a local (real analytic) system of coordinates centered on $x_{0}$ of the form $x^{1}, \dots, x^{p}, y, u^{1}, \dots, u^{s}, v^{1}, \dots, v^{r}$ so that $P$ is given in this neighborhood by the equations $y = u = v = 0$ , $R$ is given in this neighborhood by the equations $v = 0$ , and, for all $x \in P$ , the polar space $H (T_{x} P)$ is spanned by the vectors ${\partial / \partial x^{j}}_{1 \leq j \leq p} \cup {\partial / \partial y} \cup {\partial / \partial v^{ρ}}_{1 \leq ρ \leq r}$ .

Now, there exists a neighborhood $U$ of $T_{x_{0}} P$ in $G_{p} (T M)$ so that every $E \in U$ with base point $z \in M$ has a basis of the form

X_{i} (E) = (\partial / \partial x^{i} + q_{i} (E) \partial / \partial y + p_{i}^{σ} (E) \partial / \partial u^{σ} + w_{i}^{ρ} (E) \partial / \partial v^{ρ}) |_{z} .

The functions $x, y, u, v, q, p, w$ form a coordinate system on $U$ centered on $T_{x_{0}} P$ . By the definition of $H (T_{x_{0}} P)$ , there exist $s$ real analytic $(p + 1)$ -forms $κ^{1}, \dots, κ^{s}$ in $I$ with the property that

H (T_{x_{0}} P) = {v \in T_{x_{0}} M ∣ κ^{σ} (v, \partial / \partial x^{1}, \partial / \partial x^{2}, \dots, \partial / \partial x^{p}) = 0 for 1 \leq σ \leq s} .

In fact, we may even assume that $κ^{σ} (v, \partial / \partial x^{1}, \partial / \partial x^{2}, \dots, \partial / \partial x^{p}) = d u^{σ} (v)$ for $1 \leq σ \leq s$ and all $v \in T_{x_{0}} M$ . By the Kähler-regularity of $T_{x_{0}} P$ , we may assume, by shrinking $U$ if necessary, that, for all $E \in V_{p} (I) \cap U$ with base point $z \in M$ , we have

H (E) = {v \in T_{z} M ∣ κ^{σ} (v, X_{1} (E), X_{2} (E), \dots, X_{p} (E)) = 0 for 1 \leq σ \leq s} .

If we seek $v \in H (E)$ of the form

v = (a \partial / \partial y + b^{σ} \partial / \partial u^{σ} + c^{ρ} \partial / \partial v^{ρ}) |_{z},

then the $s$ equations $κ^{σ} (v, X_{1} (E), X_{2} (E), \dots, X_{p} (E)) = 0$ are, of course, linear equations for the quantities $a$ , $b$ , and $c$ of the form

A^{σ} (E) a + B_{τ}^{σ} (E) b^{τ} + C_{ρ}^{σ} (E) c^{ρ} = 0.

Again, by hypothesis, when $E = T_{x_{0}} P$ these $s$ equations are linearly independent and reduce to the equations $b^{σ} = 0$ . Thus, by shrinking $U$ if necessary, we may assume that the $s \times s$ matrix $B (E) = (B_{τ}^{σ} (E))$ is invertible for all $E \in U$ . It follows that there exist unique real analytic functions $G^{σ}$ on $U$ so that, for each $E \in U$ based at $z \in M$ , the vector $Y (E) = (\partial / \partial y + G^{σ} (E) \partial / \partial u^{σ}) |_{z}$ satisfies

κ^{σ} (Y (E), X_{1} (E), X_{2} (E), \dots, X_{p} (E)) = 0.

Since the functions $x, y, u, v, q, p, w$ form a coordinate system on $U$ centered on $T_{x_{0}} P$ , we may regard the functions $G^{σ}$ as functions of these variables.

We first show that there exists a real analytic submanifold of $R$ of the form $v = 0, u = F (x, y)$ on which the forms $κ^{σ}$ vanish. Note that the following vectors would be a basis of the tangent space to such a submanifold at the point $z (x, y) = (x, y, F (x, y), 0)$ :

X_{i} (x, y) = (\partial / \partial x^{i} + \partial_{i} F^{σ} (x, y) \partial / \partial u^{σ}) |_{z (x, y)}

Y (x, y) = (\partial / \partial y + \partial_{y} F^{σ} (x, y) \partial / \partial u^{σ}) |_{z (x, y)} .

It follows that the function $F$ would have to be a solution to the system of P.D.E. given by

(13) \partial_{y} F^{σ} = G^{σ} (x, y, F, 0, 0, \partial_{x} F, 0) .

Moreover, in order that the submanifold contain $P$ (which is given by the equations $y = u = v = 0$ ), it is necessary that the function $F$ satisfy the initial condition

(14) F (x, 0) = 0.

Conversely, if $F$ satisfies (13) and (14), then the submanifold of $R$ given by $v = 0$ and $u = F (x, y)$ will both contain $P$ and be an integral of the set of forms ${κ^{σ}}_{1 \leq σ \leq s}$ .

By the Cauchy–Kowalevski theorem, there exists a unique real analytic solution $F$ of (13) and (14). We let $X \subset R$ denote the (unique) submanifold of dimension $p + 1$ constructed by this method. Replacing the functions $u$ in our coordinate system by the functions $u - F (x, y)$ will not disturb any of our normalizations so far and allows us to suppose, as we shall for the remainder of the proof, that $X$ is described by the equations $u = v = 0$ .

We must now show that $X$ is an integral manifold of $I$ . We have already seen that $X$ is the unique connected real analytic submanifold of dimension $p + 1$ which satisfies $P \subset X \subset R$ and is an integral of the forms ${κ^{σ}}_{1 \leq σ \leq s}$ . We now show that all of the $p$ -forms in $I$ vanish on $X$ .

Again using the Kähler-regularity of $T_{x_{0}} P$ , let $β^{1}, \dots, β^{a}$ be a set of real analytic $p$ -forms in $I$ so that the functions $f^{c} (E) = β^{c} (X_{1} (E), \dots, X_{p} (E))$ for $1 \leq c \leq a$ have linearly independent differentials on $U$ and have the locus $V_{p} (I) \cap U$ as their set of common zeros. (We may have to shrink $U$ once more to do this.) Since $T_{x_{0}} X$ lies in $V_{p + 1} (I)$ by construction, Proposition 1.8 shows that $T_{x_{0}} X$ is Kähler-ordinary. In fact, the proof of Proposition 1.8 shows that the $(p + 1)$ -forms ${β^{c} \land d y}_{1 \leq c \leq a} \cup {κ^{σ}}_{1 \leq σ \leq s}$ have $V_{p + 1} (I) \cap U^{+}$ as their set of ordinary common zeros in some neighborhood $U^{+}$ of $T_{x_{0}} X$ in $G_{p + 1} (T M)$ . Thus, in order to show that $X$ is an integral manifold of $I$ , it suffices to show that the forms ${β^{c} \land d y}_{1 \leq c \leq a}$ vanish on $X$ .
Shrinking $U^{+}$ if necessary, we may suppose that every $E^{+} \in U^{+}$ has a basis

X_{1} (E^{+}), X_{2} (E^{+}), \dots, X_{p} (E^{+}), Y (E^{+})

that is dual to the basis of 1-forms $d x^{1}, d x^{2}, \dots, d x^{p}, d y$ . If we set

B^{c} (E^{+}) = β^{c} \land d y (X_{1} (E^{+}), X_{2} (E^{+}), \dots, X_{p} (E^{+}), Y (E^{+}))

K^{σ} (E^{+}) = κ^{σ} (X_{1} (E^{+}), X_{2} (E^{+}), \dots, X_{p} (E^{+}), Y (E^{+})),

then we have

V_{p + 1} (I) \cap U^{+} = {E^{+} \in U^{+} ∣ B^{c} (E^{+}) = K^{σ} (E^{+}) = 0} .

Since $I$ is an ideal, the forms $β^{c} \land d x^{i}$ are also in $I$ and hence vanish on $V_{p + 1} (I) \cap U^{+}$ . Thus, if we set

B^{c i} (E^{+}) = β^{c} \land d x^{i} (X_{1} (E^{+}), X_{2} (E^{+}), \dots, X_{p} (E^{+}), Y (E^{+})),

then the functions $B^{c i}$ are in the ideal generated by the functions $B^{c}$ and $K^{σ}$ . It follows that there exist real analytic functions $A$ and $L$ on $U^{+}$ so that

B^{c i} = A_{b}^{c i} B^{b} + L_{σ}^{c i} K^{σ} .

Since $K^{σ} (T_{z} X) = 0$ for all $z \in X$ by construction, it follows that

B^{c i} (T_{z} X) = A_{b}^{c i} (T_{z} X) B^{b} (T_{z} X)

for all $z \in X$ .
Since $I$ is differentially closed, the forms $d β^{c}$ are in $I$ . Thus, if we set

D^{c} (E^{+}) = d β^{c} (X_{1} (E^{+}), X_{2} (E^{+}), \dots, X_{p} (E^{+}), Y (E^{+})),

there must exist functions $G$ and $H$ on $U^{+}$ so that

D^{c} = G_{b}^{c} B^{b} + H_{σ}^{c} K^{σ} .

Again, since $K^{σ} (T_{z} X) = 0$ , we must have

D^{c} (T_{z} X) = G_{b}^{c} (T_{z} X) B^{b} (T_{z} X)

for all $z \in X$ .

Now, if we restrict the forms $β^{c}$ to $X$ , then we have an expansion of the form

\begin{aligned} β^{c} |_{X} = & B^{c} (x, y) d x^{1} \land \dots \land d x^{p} \\ + \sum_{i} (- 1)^{p - i + 1} B^{c i} (x, y) d x^{1} \land \dots \land d x^{i - 1} \land d x^{i + 1} \land \dots \land d x^{p} \land d y, \end{aligned}

where, for $z = (x, y, 0, 0) \in X$ , we have set $B^{c} (x, y) = B^{c} (T_{z} X)$ and $B^{c i} (x, y) = B^{c i} (T_{z} X)$ . We also have the formula

\begin{aligned} d β^{c} |_{X} & = (- 1)^{p} (\partial_{y} B^{c} (x, y) + \sum_{i} \partial_{i} B^{c i} (x, y)) d x^{1} \land \dots \land d x^{p} \land d y \\ = D^{c} (x, y) d x^{1} \land \dots \land d x^{p} \land d y, \end{aligned}

where we have written $D^{c} (x, y) = D^{c} (T_{z} X)$ for $z$ as before. Using the formulas

\begin{aligned} D^{c} (x, y) & = G_{b}^{c} (x, y) B^{b} (x, y) \\ B^{c i} (x, y) & = A_{b}^{c i} (x, y) B^{b} (x, y), \end{aligned}

we see that the functions $B^{c} (x, y)$ satisfy a linear system of P.D.E. of the form

\partial_{y} B^{c} (x, y) = {\tilde{A}}_{b}^{c i} (x, y) \partial_{i} B^{b} (x, y) + {\tilde{G}}_{b}^{c} (x, y) B^{b} (x, y)

for some functions $\tilde{A}$ and $\tilde{G}$ on $X$ . Moreover, since $T_{z} X$ is an integral element of $I$ when $y = 0$ , we have the initial conditions $B^{c} (x, 0) = 0$ . By the uniqueness part of the Cauchy–Kowalevski theorem and the fact that all of the functions involved are real-analytic, it follows that the functions $B^{c} (x, y)$ must vanish identically. In turn, this implies that the forms $β^{c}$ vanish on $X$ . Hence $X$ is an integral manifold of $I$ , as we wished to show. Since we have already established uniqueness, we are done. $◻$