Example

If F has rank 1 at every point, F is integrable, and defines on X a pure foliation of dimension 1. Suppose in addition that $K = \mathbf{R}$, that X is separated, and that F has as a frame a vector field $\xi$ everywhere nonzero. Then, for every $x \in X$, the maximal connected leaf containing $x$ is the image of the maximal integral arc of $\xi$ with origin $x$ (9.1.3).

9.3.3 (« Integrability Criteria »). The following conditions are equivalent:

(i) The vector subbundle F of $T(X)$ is integrable.

(ii) For every $x \in X$, there exists an integral submanifold Z of class $C^s$ of F such that $x \in Z$ and $T_x(Z) = F_x$.

(iii) Whatever the open set U of X and the vector fields $\xi$ and $\eta$ belonging to $\mathscr{J}_F^s(U)$ (7.4.1), one has $[\xi, \eta] \in \mathscr{J}_F^{s-1}(U)$.

Let $(\xi_i)_{i \in I}$ be a family of sections of F of class $C^s$ such that, for every $x \in X$, the set of the $\xi_i(x)$ is a total subset of the Banach space $F_x$ (EVT, I, § 2, No. 1). The conditions (i), (ii), (iii) are then equivalent to:

(iv) for every pair $(i, j)$ of elements of I, and every $x \in X$, one has $[\xi_i, \xi_j](x) \in F_x$.

When I is finite, and the family $(\xi_i)$ is a frame of F, the preceding conditions are also equivalent to:

(v) there exists a family $(c_{ij}^k)_{(i, j, k) \in I \times I \times I}$ of functions on X with values in K such that $[\xi_i, \xi_j] = \sum_k c_{ij}^k \xi_k$ for all $i, j$ in I.

(If this is the case, the functions $c_{i,j}^k$ are of class $C^{s-1}$.)

9.3.4. Let $s' \in \mathbf{N}_K$, with $s' \leq s$, and let $F'$ be the vector bundle of class $C^{s'}$ deduced from F by weakening of structure (8.7.1). In order that F be integrable, it is necessary and sufficient that $F'$ be so. In this case, the integral foliation of $F'$ is deduced from that of F by weakening of structure.

9.3.5. Let E be a Banach space and $p$ an integer $\geq 0$. For every $x \in X$, let us denote

N(p, E)_x the vector subspace of Alt^p(T_x(X); E) formed by the elements u such that $u(v_1, \ldots, v_p) = 0$ for $v_1, \ldots, v_p$ in $F_x$. The spaces N(p, E)_x, for $x \in X$, are the fibres of a vector subbundle of class C^s of Alt^p(T(X); E). Let us denote it by N(p, E). If F is integrable, we have:

(vi) for every open U of X and every form $\omega \in \mathscr{S}_{N(p, E)}^s(U)$, one has $d\omega \in \mathscr{S}_{N(p+1, E)}^{s-1}(U)$.

Conversely, if (vi) is satisfied for $p = 1$ and for every Banach space E, the bundle F is integrable; when K = R or C, it is even enough to verify (vi) for $p = 1$ and E = K.

Suppose that the dual of T(X)/F admits a frame $(\omega_1, \ldots, \omega_n)$. The integrability of F is then equivalent to the following condition:

(vii) $d\omega_i \wedge \omega_1 \wedge \cdots \wedge \omega_n = 0$ for $1 \leq i \leq n$.

If in addition there exists a vector subbundle G of class C^s of T(X) such that T(X) is the direct sum of F and G, condition (vii) is equivalent to:

(viii) there exist differential forms $\alpha_i^j$ (i, j in I) of degree 1 on X, of class C^{s-1}, such that $d\omega_i = \sum_j \alpha_i^j \wedge \omega_j$ for all $i \in I$.

9.3.6. Let L be a Banach space and let $\omega$ be a differential form of degree 1 on X with values in L, of class C^s. We make the following two hypotheses:

a) for every $x \in X$, $\omega_x$ is a surjective homomorphism from $T_x(X)$ into L, and the kernel of $\omega_x$ is $F_x$; b) there exists a vector subbundle G of T(X) such that T(X) is the direct sum of F and G.

The integrability of F is then equivalent to:

(ix) there exists a differential form $\alpha$ of degree 1 on X with values in End(L) such that $d\omega = \alpha \wedge \omega$, the exterior product being defined by the canonical pairing End(L) $\times$ L $\to$ L (7.8.2 and 8.3.2).

9.3.7 (« Equations in total differentials »). Suppose that X is the product of two manifolds A and B of class C^r; denote by $p_1 : X \to A$ and $p_2 : X \to B$ the two projections. Let f be a morphism of class C^s from $p_1^*T(A)$ into $p_2^*T(B)$. The graphs of the mappings $f_{(a, b)} : T_a(A) \to T_b(B)$ are the fibres of a vector subbundle of class C^s of T(X); denote it by F^f.

Let A’ be an open subset of A, and let $\varphi : A' \to B$ be a morphism of class C^k ($k \in \mathbf{N}_K, k \leq r$). We say that $\varphi$ is an integral of f if, for every $a \in A'$, one has $T_a(\varphi) = f_{a, \varphi(a)}$; this is equivalent to saying that the mapping $a \mapsto (a, \varphi(a))$ from A’ into X is an integral of F^f (9.3.1). Such a mapping is of class C^{s+1}. If $\varphi_1$ and $\varphi_2$ are two integrals of f and take the same value at a point $a \in A$, they coincide in a neighbourhood of a.

More generally, let Z be a manifold of class C^k, let A’ be an open subset of A, let $a \in A'$ and let $\Phi_1, \Phi_2$ be morphisms of class C^k from Z $\times$ A’ into B. Suppose that

Φ₁ and Φ₂ coincide on Z × {a} and that, for every z ∈ Z, the morphisms

are integrals of f. Then Φ₁ and Φ₂ coincide in a neighbourhood of Z × {a}.

Suppose that F^f is integrable. Let Z be a manifold of class C^k (k ∈ N_K, k ≤ s), let (z₀, a₀) be a point of Z × A, and let ρ be a morphism of class C^k from Z into B. There exists an open neighbourhood Z' × A' of (z₀, a₀) in Z × A and a morphism Φ : Z' × A' → B of class C^k such that, for every z ∈ Z', the mapping a ↦ Φ(z, a) from A' into B has f as its tangent mapping and takes the value ρ(z) at the point a₀.

9.3.8. Retain the hypotheses and notations of 9.3.7 and suppose that A (resp. B) is an open submanifold of a Banach space E (resp. M). The mapping f is then identified with a morphism of class C^s from X = A × B into the Banach space $\mathscr{L}(E; M)$. Denote by D₁f (resp. D₂f) the first (resp. second) partial derivative of f (1.6.2); this is a morphism of class C^{s-1} from X into $\mathscr{L}(E; \mathscr{L}(E; M))$ (resp. into $\mathscr{L}(M; \mathscr{L}(E; M))$), a space which is identified in the obvious way with $\mathscr{L}_2(E; M)$ (resp. with $\mathscr{L}(M, E; M)$). For F to be integrable, it is necessary and sufficient that, for every x ∈ X, the bilinear mapping

from E × E into M be symmetric. Under this condition, if φ is an integral of f defined in an open subset A' of A, the second derivative of φ at a point a of A' is $\Delta_{(a, \varphi(a))}$.

If E = K^n, and if we denote by (x¹, ..., xⁿ) the coordinate functions on K^n, the mapping f is defined by a family (f₁, ..., fₙ) of mappings from A × B into M, and the integrability condition is written:

whatever the integers i, j in {1, n}. A mapping φ from an open subset A' of A into B is an integral of f if and only if one has in other words, if one has