§ 8. The differential calculus of order 1

Starting from No. 8.3, we suppose

— either that $K$ is of characteristic zero,

— or that the varieties under consideration are locally of finite dimension.

In the latter case, the expressions “Banach space, vector bundle, vector functor” mean respectively “finite-dimensional vector space, vector bundle of finite rank, vector functor in finite dimension with values of finite dimension”.

8.1. Tangent bundle

8.1.1. — Let $r\in \mathbf{N}_K$ and let $X$ be a $K$-variety of class $C^r$. We denote by $T(X)$ the sum set of the tangent spaces $T_x(X)$ (5.5.1) for $x\in X$, and by $\pi$ the canonical mapping of $T(X)$ onto $X$. Let $c=(U,\varphi,E)$ be a chart of the variety $X$; for $x\in U$ and $t\in T_x(X)$, put $\zeta_c(x,t)=(x,\theta_c^{-1}(t))$, the mapping $\theta_c$ being that defined in 5.5.1. The triplet $c'=(U,\zeta_c,E)$ is a vector chart (7.1.1) of $T(X)$ (endowed with the mapping $\pi:T(X)\longrightarrow X$). There exists on $T(X)$ one and only one structure of vector bundle with base $X$ and of class $C^{r-1}$ such that, for every chart $c$ of $X$, the corresponding vector chart $c'$ is a vector chart of $T(X)$[^1]. Endowed with this structure, $T(X)$ is called the tangent bundle of $X$. In particular, $T(X)$ is endowed with a structure of variety of class $C^{r-1}$ and the triplet $(T(X),X,\pi)$ is a fibration (7.1.4).

Let $U$ be an open subset of a Banach space $E$ and let $i$ be the canonical injection of $U$ into $E$. The vector chart $c'=(U,\zeta_c,E)$ associated with the chart $c=(U,i,E)$ of $U$ defines an isomorphism of $T(U)$ onto the trivial bundle $E_U$ (7.1.5); these two bundles are identified by means of this isomorphism.

If $X$ is a purely variety of type $E$ (5.1.7), $T(X)$ is a purely variety of type $E\times E$.

8.1.2. — Let $f:X\longrightarrow Y$ be a morphism of varieties of class $C^r$. We define an $f$-morphism $T(f):T(X)\longrightarrow T(Y)$ of vector bundles of class $C^{r-1}$ by

If $f=\mathrm{Id}_X$, one has $T(f)=\mathrm{Id}_{T(X)}$. For $f:X\longrightarrow Y$ and $g:Y\longrightarrow Z$, one has

8.1.3. — Let $f:X\longrightarrow Y$ be a morphism of manifolds of class $C^r$, and let $f':T(X)\longrightarrow f^*T(Y)$

be the unique X-morphism such that T(f) = g ∘ f', where g is the canonical morphism f*T(Y) → T(Y) (cf. 7.2.4).

For f to be an immersion (resp. a submersion), it is necessary and sufficient that 0 → T(X) $\xrightarrow{f'} f^*T(Y)$ (resp. T(X) $\xrightarrow{f'} f^*T(Y) \to 0$) be a locally direct exact sequence (7.5.6). If f is an immersion, the cokernel bundle of f' is called the normal or transverse bundle of f. If f is a submersion, the kernel bundle of f is called the tangent bundle to the fibres of f, or the relative tangent bundle of X over Y, and is denoted T(X/Y). Its fibre T_x(X/Y) at x ∈ X is the tangent space at x to the submanifold $f^{-1}(f(x))$ of X. The sequence

where i is the canonical injection, is exact.

For f to be étale, it is necessary and sufficient that f' be an isomorphism.

When K has characteristic 0, for f to be a subimmersion, it is necessary and sufficient that f' be locally direct.

8.1.4. Let X_1 and X_2 be two manifolds. We have T(X_1 × X_2) = p_1^*T(X_1) ⊕ p_2^*T(X_2), where p_1 and p_2 are the canonical projections (5.6.3).

8.1.5. Let n be an integer $\geqslant 0$; the tangent bundle T(K^n) of K^n is identified with the trivial bundle with fibre K^n (8.1.1). The constant sections defined by the elements of the canonical basis of K^n form a frame of T(K^n).

If X is a manifold and (u^1, ..., u^n) a system of coordinates of X in an open set U of X (5.1.10), the preceding frame defines by transport of structure a frame of T(X) on U; it is called the tangent frame defined by (u^1, ..., u^n) and is denoted (\partial/\partial u^1, ..., \partial/\partial u^n) (cf. 5.5.8).

8.2. Vector fields

Let X be a manifold of class C^r, and let U be an open set of X.

8.2.1. A section (not necessarily continuous, cf. Notations and Conventions) of the tangent bundle T(X) above U is called a vector field on U.

8.2.2. The dual bundle T'(X) (7.7.3) of T(X) is called the cotangent bundle of X. Its sections are called covector fields. If f is a function of class C^k on X (with 1 ≤ k ≤ r), with values in K, its differential df : x ↦ d_x f (5.5.6) is a covector field of class C^{k-1}. More generally, if f is a function of class C^k on X (1 ≤ k ≤ r), with values in a Banach space E, its differential df is a section of class C^{k-1} of the bundle $\mathscr{L}(T(X); E_X)$.

8.2.3. Let $\xi$ be a vector field of class C^{r-1} on X, and let $f \in \mathscr{C}^r(X; F)$ be a function of class C^r with values in a Banach space F. We denote by $\langle \xi, df \rangle$, or D_\xi(f), or again $\xi(f)$ the function $x \mapsto d_x f(\xi(x))$; it is an element of $\mathscr{C}^{r-1}(X; F)$. For fixed f, the mapping $\xi \mapsto D_\xi(f)$ is $\mathscr{C}^{r-1}(X)$-linear. If $D_\xi f = 0$ for every function f of class C^r with values in a Banach space, defined in an open set of X, we have $\xi = 0$.

Let E, F, G be Banach spaces and let (u, v) ↦ u . v be a continuous bilinear mapping of $E \times F$ into $G$. Let $f \in \mathscr{C}^r(X; E)$, $g \in \mathscr{C}^r(X; F)$ and let $f.g \in \mathscr{C}^r(X; G)$ be their product. If $\xi$ is a vector field of class $C^{r-1}$ on $X$, we have

8.2.4. Suppose that $X$ is locally of finite dimension and that one of the following two hypotheses is satisfied:

(i) $r = \infty$ and $X$ is separated; (ii) $r = \omega$ and $X$ is isomorphic to an open polydisc of $K^n$.

Then, for every derivation $D$ of the algebra $\mathscr{C}^r(X)$, there exists one and only one vector field $\xi$ of class $C^r$ such that $D = D_\xi$.

8.2.5. Let $(u^1, \ldots, u^n)$ be a system of coordinates of $X$ in $U$, and $\xi$ a vector field on $U$. The $i$-th coordinate of $\xi$ in the tangent frame (8.1.5) defined by $(u^1, \ldots, u^n)$ is $D_\xi(u^i)$. In other words, we have

8.2.6. Let $\varphi : X \to Y$ be a morphism of manifolds of class $C^r$, $\xi$ a vector field on $X$ and $\eta$ a vector field on $Y$. We say that $\xi$ is $\varphi$-*related* to $\eta$ if, for every $x \in X$, we have $\eta(\varphi(x)) = T_x(\varphi)(\xi(x))$. Suppose this is so, and that $\xi$ and $\eta$ are of class $C^{r-1}$. Then, for every Banach space $E$, the diagram

where $\varphi^*(f) = f \circ \varphi$, is commutative.

When $\varphi$ is étale, for every vector field $\eta$ on $Y$, there exists one and only one vector field on $X$ which is $\varphi$-*related* to $\eta$; it is denoted by $\varphi^*\eta$.

8.2.7. Let $\varphi : X \to Y$ be a morphism of manifolds of class $C^r$. If $\omega$ is a covector field$^{(1)}$ on $Y$, we define a covector field $\varphi^*\omega$ on $X$ by

$^1$ The reader should take care not to confuse this use of the letter $\omega$ with that defined in Notations and Conventions.

8.3. Differential forms, exterior differentiation

Let us recall that, in this number and the following ones, it is assumed either that K is of characteristic 0, or that the manifolds considered are locally of finite dimension.

8.3.1. Let X be a manifold of class C' and F a vector bundle of class C^k based on X, with k + 1 \in \mathbf{N}_K and k + 1 \leq r. Let p be an integer $\geq 0$ and let Alt^p(T(X); F) be the vector bundle of class C^k of alternating p-linear mappings of T(X) into F (7.8.1 and Errata to No. 7.8). A section of this bundle over an open set U of X is called an alternating differential form (or simply differential form) of degree p on U, with values in F. Those of these sections which are of class C^k form a $\mathscr{C}^k(U)$-module, denoted ^k\Omega^p(U; F). If E is a Banach space, we write ^k\Omega^p(U; E) instead of ^k\Omega^p(U; E_X) and speak of differential forms with values in E. In this notation, k (resp. E) is sometimes omitted when it is equal to r − 1 (resp. K).

A differential form of degree 0 (resp. 1) with scalar values is a function (resp. a covector field).

Let $\varphi : Y \to X$ be a morphism of manifolds of class C', and let $\omega \in ^k\Omega^p(X; F). There then exists one and only one form$\varphi^*\omega \in ^k\Omega^p(Y; \varphi^*F) such that

(1)

for every y \in Y and every family v_1, \ldots, v_p of elements of T_y(Y).

If $\psi : Z \to Y$ is a morphism of manifolds of class C', we have

(2)

When Y is a submanifold of X and $\varphi$ is the canonical injection, we sometimes write $\omega|Y$ instead of $\varphi^*\omega$ and say that $\omega|Y$ is induced on Y by $\omega$.

8.3.2. The definitions and results of No. 7.8 apply to differential forms. In particular, a pairing of vector bundles F' \times_X F'' \to F gives rise to an exterior product ^k\Omega^{p'}(U; F) \times ^k\Omega^{p''}(U; F'') \to ^k\Omega^p(U; F), where p = p' + p'' (7.8.2); it is denoted by $(\omega', \omega'') \mapsto \omega' \wedge \omega''$. The direct sum of the ^k\Omega^p(U; K) for p \geq 0 is thus endowed with a structure of associative and alternating graded algebra (A, III, p. 53).

Let $\xi$ be a vector field on X and $\omega$ a differential form of degree p \geq 1 on X, with values in a vector bundle F; the interior product of $\xi$ and $\omega$ is the differential form i(\xi, \omega) of degree p − 1 on X, with values in F, defined by

(3)

for every x \in X and every family v_1, \ldots, v_{p-1} of elements of T_x(X). If $\omega$ is a differential form of degree 0, we put i(\xi, \omega) = 0. We also write i(\xi)\omega or i_\xi\omega instead of i(\xi, \omega). When F = K_X, the definition given above coincides with that of 7.8.4.

Let F' \times_X F'' \to F be a pairing of vector bundles. For $\omega' \in ^k\Omega^{p'}(U; F') and$\omega'' \in ^k\Omega^{p''}(U; F''), we have

(4)

8.3.3. Let $(u^1, \ldots, u^n)$ be a system of coordinates in the open set $U$ of $X$. Every element $\omega$ of $^{k}\Omega^p(U; F)$ is written in a unique way

8.3.4. Let $p$ be an integer $\geqslant 0$ and $r$ an element of $N_K$. Let $E$ and $F$ be Banach spaces, $U$ an open set of $E$ and $\alpha \in {}^r\Omega^p(U; F)$. Let $\tilde{\alpha} \in C^r(U; Alt^p(E; F))$ be the element corresponding to $\alpha$. If $x \in U$, the differential $d_x \tilde{\alpha}$ of $\tilde{\alpha}$ at $x$ (5.5.6) is an element of $\mathscr{L}(E; Alt^p(E; F))$; if $t \in E$, we have $(d_x \tilde{\alpha})(t) \in Alt^p(E; F)$, and, if $t_1, \ldots, t_p$ are in $E$, we have $(d_x \tilde{\alpha})(t)(t_1, \ldots, t_p) \in F$. There exists one and only one element $d\alpha$ of ${}^{r-1}\Omega^{p+1}(U; F)$ such that one has

8.3.5. Let $p$ be an integer $\geqslant 0$ and $r$ an element of $N_K$. Let $X$ be a variety of class $C^{r+1}$, $F$ a Banach space, and $\omega \in {}^r\Omega^p(X; F)$. There exists one and only one differential form $\pi \in {}^{r-1}\Omega^{p+1}(X; F)$ such that, for every chart $c = (U, \varphi, E)$ of $X$, if $\omega_c$ is the differential form on $\varphi(U)$ such that $\omega|U = \varphi^*(\omega_c)$, one has $\pi_c = d\omega_c$, where $d\omega_c$ is defined by formula (5) of 8.3.4.

The differential form $\pi$ is called the exterior differential of $\omega$; it is denoted $d\omega$. It has the following properties:

(6) If $\psi : Y \to X$ is a morphism of manifolds of class $C^{r+1}$, and if $\omega \in {}^r\Omega^p(X; F)$, one has $\psi^*(d\omega) = d(\psi^*\omega)$; in particular, $d$ commutes with the operation of restriction to a submanifold. (7) For $p = 0$, the mapping $d : {}^r\Omega^0(X; F) \to {}^{r-1}\Omega^1(X; F)$ coincides with that defined in 8.2.2. (8) If $r \geqslant 2$ and if $\omega \in {}^r\Omega^p(X; F)$, one has $d(d\omega) = 0$.\footnote{The sign $\wedge$ indicates that the symbol above which it is placed is to be omitted (cf. 7.8.4).} (9) For every continuous linear mapping $u : F \to F'$ of Banach spaces, one has $d(u(\omega)) = u(d(\omega))$ for $\omega \in {}^r\Omega^p(U; F)$. (10) For every continuous bilinear mapping $F' \times F'' \to F$ of Banach spaces, one has

8.3.6. Let $(u^1, \ldots, u^n)$ be a system of coordinates of $X$ in $U$. Let

1 The sign $\wedge$ indicates that the symbol above which it is placed is to be omitted (cf. 7.8.4). 2 If $\omega \in {}^1\Omega^p(X; F)$ and if $d\omega \in {}^1\Omega^{p+1}(X; F)$, one still has $d(d\omega) = 0$.

a differential form on U (with $f_{i_1, ..., i_p} \in \mathscr{C}^r(U; F)$). One then has

8.3.7. Suppose that K is of characteristic zero. Let $\omega \in {}^r\Omega^p(X; F)$ be a differential form of degree $p \geq 1$ such that $d\omega = 0$. For every $x \in X$, there exists an open neighbourhood U of $x$ and a differential form $\pi \in {}^r\Omega^{p-1}(U; F)$ such that $d\pi = \omega$ on U.

8.4. Infinitesimal transformations

Let $r \in \mathbf{N}_K$ and X be a manifold of class $C^{r+1}$.

8.4. Infinitesimal transformations

8.4.1. Let $\tau$ be a vector functor for isomorphisms (7.6.6), of class $C^r$. If E is a Banach space, denote by $\mathbf{GL}(E)$ the open subset of $\mathscr{L}(E; E)$ consisting of the automorphisms of E. Denote by $\tau'_E$ the tangent linear mapping at the identity element $\mathrm{Id}_E$ of the morphism $\tau : \mathbf{GL}(E) \to \mathbf{GL}(\tau(E))$; it is a continuous linear mapping of $\mathscr{L}(E; E)$ into $\mathscr{L}(\tau(E); \tau(E))$.

8.4.2. Let E be a Banach space and put $F = \tau(E)$. Let U be an open subset of E, $\xi$ a vector field on U of class $C^r$, $\tilde{\xi} : U \to E$ the corresponding mapping (obtained by identifying T(U) and $U \times E$ (8.1.1)), and $f : U \to F$ a mapping of class $C^r$. For $x \in U$, define an element $(\theta_{\xi}.f)(x)$ of F by the formula

(Let us remark that on the one hand $\tilde{\xi}(x) \in E$ and $d_x f \in \mathscr{L}(E; F)$, whence $d_x f(\tilde{\xi}(x)) \in F$, and on the other hand, $d_x \tilde{\xi} \in \mathscr{L}(E; E)$, $\tau'_E(d_x \tilde{\xi}) \in \mathscr{L}(F; F)$ and $f(x) \in F$, whence $\tau'_E(d_x \tilde{\xi})(f(x)) \in F$. The function $\theta_{\xi}.f : x \mapsto (\theta_{\xi}.f)(x)$ is of class $C^{r-1}$ in U.

8.4.3. Let F denote the vector bundle $\tau(T(X))$ (7.6.2 and 7.6.6). Let $\xi$ be a vector field on X and $f$ a section of F, both of class $C^r$. Let $x$ be a point of X and $c = (U, \varphi, E)$ a chart of X at $x$. Let $c' = (U, \zeta_c, E)$ be the corresponding vector chart of T(X) (8.1.1) and $\tau(c') = (U, \psi_c, \tau(E))$ the corresponding vector chart of F (7.6.2). Put $x_c = \varphi(x)$; let $\xi_c$ be the mapping of $\varphi(U)$ into E such that $\zeta_c(\xi(u)) = (u, \xi_c(\varphi(u)))$ and $f_c$ the mapping of $\varphi(U)$ into $\tau(E)$ such that $\psi_c(f(u)) = (u, f_c(\varphi(u)))$ for all $u \in U$. Let $a_c$ be the element $(\theta_{\xi_c}.f_c)(x_c)$ of $\tau(E)$ defined in 8.4.2. There exists one and only one element $a$ of $F_x$ such that $\psi_c(a) = (x, a_c)$ for every chart $c$ of X at $x$. We denote it by $(\theta_{\xi}.f)(x)$. It depends only on the germs of $\xi$ and $f$ at $x$ (and even only on their jet of order 1 (12.1.2)).

The mapping $\theta_{\xi}.f : x \mapsto (\theta_{\xi}.f)(x)$ is a section of class $C^{r-1}$ of F; it depends K-bilinearly on the pair $(\xi, f)$.

8.4.4. Let us retain the preceding hypotheses and notations. There exists (7.6.4) one and only one morphism of vector bundles $\tau'$ from $\mathscr{L}(T(X); T(X))$ into $\mathscr{L}(F; F)$ such that, for every $x \in X$, the restriction of $\tau'$ to the fibre $\mathscr{L}(T(X); T(X))_x = \mathscr{L}(T_x(X); T_x(X))$ is equal to $\tau'_{T(x)}$ (8.4.1). If $g$ is a function of class $C^r$ in $X$, with values in $K$, we have

8.4.5. Let $I$ be an open set in $K$ containing $0$, $U$ an open set in $X$ and $\varphi$ a mapping of class $C^r$ from $I \times U$ into $X$. For $(t, x) \in I \times U$, denote by $\varepsilon_{t,x}$ the vector $(1, 0) \in K \times T_x(X) = T_{(t,x)}(I \times U)$. For $t \in I$, denote by $\varphi_t$ the mapping from $U$ into $X$ defined by $\varphi_t(x) = \varphi(t, x)$ ($x \in U$). We suppose that the following three conditions are satisfied:

a) $\varphi_0$ is the canonical injection of $U$ into $X$; b) for every $t \in T$, the mapping $\varphi_t$ is an isomorphism of $C^r$ class manifolds of $U$ onto an open set of $X$; c) the section $(t, x) \mapsto T_{(t, x)}(\varphi)(\varepsilon_{t, x})$ of the vector bundle $\varphi^*T(X)$ is of class $C^r$. Moreover, put: d) $\xi(x) = T_{(0, x)}(\varphi)(\varepsilon_{0, x})$ for every $x \in U$.

The mapping $\xi : x \mapsto \xi(x)$ is a vector field of class $C^r$ on $U$. We call it the initial (or starting) vector field of $\varphi$.

Let $\tau$ be a vector functor for isomorphisms, of class $C^r$; put $F = \tau(T(X))$ and let $f$ be a section of class $C^r$ of $F$ over $X$. For $x \in U$ and $t \in I$, put $\varphi_t^*f(x) = \tau(T_x(\varphi_t)^{-1})(f(\varphi_t(x)))$; this is an element of $F_x$. The mapping $t \mapsto \varphi_t^*f(x)$ of $I$ into $F_x$ is of class $C^{r-1}$ for all $x \in U$ and one has

8.4.6. Given a vector field $\xi$ of class $C^r$ on $X$ and a point $x_0$ of $X$, there exists an open set $I$ of $K$, containing $0$, an open set $U$ of $X$ containing $x_0$ and a mapping $\varphi : I \times U \to X$ of class $C^r$ satisfying conditions $a), b)$ and c) of 8.4.5 and such that the initial field of $\varphi$ is the restriction $\xi|U$ of $\xi$ to $U$.

8.4.7. Example. — Let $F$ be a Banach space and let $p$ be an integer $\geqslant 0$. If $V$ is a Banach space, put $\alpha_p(V) = \mathrm{Alt}^p(V; F)$; if $u$ is an isomorphism of $V$ onto a Banach space $V'$, denote by $\alpha_p(u)$ the isomorphism of $\alpha_p(V)$ onto $\alpha_p(V')$ deduced from $u$ by transport of structure. We thus obtain a vector functor for isomorphisms, denoted by $\alpha_p$ (7.8.1 and Errata to No. 7.8). We can apply to it what precedes; we have

and, when $r \geq 2$,

When $p = 0$, the functor $\alpha_p$ is the constant vector functor defined by $F$; the bundle $\alpha_0(T(X))$ is identified with the trivial bundle $F_X$, and the sections of this bundle are identified with the functions on $X$ with values in $F$. If $\xi$ is a vector field of class $C^r$ on $X$ and if $f \in \mathscr{C}^r(X; F)$, we have

it is an element of $\mathscr{C}^{r-1}(X; F)$.

8.4.8. Let $\tau$ and $\tau_1$ be two vector functors for isomorphisms. A morphism of vector functors $h : \tau_1 \to \tau$ defines a morphism of vector bundles, also denoted by $h$, from $\tau_1(T(X))$ into $\tau(T(X))$ (7.6.4), and makes correspond to a section $f$ of $\tau_1(T(X))$ a section $h(f)$ of $\tau(T(X))$. If $\xi$ is a vector field on $X$, and if $f$ and $\xi$ are of class $C^r$, we have $\theta_\xi \cdot h(f) = h(\theta_\xi \cdot f)$.

More generally, let $\tau, \tau_1, \ldots, \tau_d$ be vector functors for isomorphisms and let $h : (\tau_1, \ldots, \tau_d) \to \tau$ be a $d$-linear morphism (7.6.4). Let $f_i \in \mathscr{S}_{\tau_i(T(X))}^r(X)$ (for $i = 1, \ldots, d$), and let $\xi$ be a vector field of class $C^r$ on $X$. We have

For example, if $f$ is a section of $\tau(T(X))$ and $g$ a function on $X$, with values in $K$, both of class $C^r$, we have

If $\omega_1$ and $\omega_2$ are two differential forms of class $C^r$, with values in Banach spaces $F_1$ and $F_2$ respectively, we have

the exterior product being taken with respect to a continuous bilinear mapping of $F_1 \times F_2$ into a Banach space $F$.

8.4.9. Let $\tau$ be a vector functor for isomorphisms, contravariant (8.2.7), of class $C^r$, $\varphi : X \to Y$ a morphism of manifolds of class $C^{r+1}$, $\xi$ (resp. $\eta$) a vector field of class $C^r$ on $X$ (resp. $Y$) such that $\xi$ is $\varphi$-related to $\eta$ (8.2.6). For every section $f$ of $\tau(T(Y))$, of class $C^r$, we have

8.5. The bracket

Let $r \in \mathbf{N}_k$ and let $X$ be a manifold of class $C^{r+1}$.

8.5.1. Apply the definitions of No. 8.4 taking for $\tau$ the identity functor: $\tau(V) = V$ for every Banach space $V$ and $\tau(u) = u$ for every isomorphism $u$ of Banach spaces. We then have $\tau(T(X)) = T(X)$. If $\xi$ and $\eta$ are two vector fields of class $C^r$ on $X$, we put

(1)