§ 5. Manifolds

5.1. Charts and atlases. Manifolds

5.1.1. Let X be a set. A chart of X is a triple c = (U, φ, E), where U is a subset of X, E a Banach space and φ a bijection of U onto an open subset of E. We say that U is the domain of the chart c. If E is of finite dimension n, we say that c is of dimension n. Otherwise, we set dim c = +∞.

5.1.2. We say that two charts c = (U, φ, E) and c' = (U', φ', E') of X are C^r-compatible (or simply compatible when there can be no ambiguity about r) when the following conditions are satisfied: (a) φ(U ∩ U') (resp. φ'(U ∩ U')) is open in E (resp. E'); (b) the mapping φ ∘ φ'^{-1} (resp. φ' ∘ φ^{-1}) of φ'(U ∩ U') onto φ(U ∩ U') (resp. of φ(U ∩ U') onto φ'(U ∩ U')) is of class C^r (cf. 2.3.1, 3.2.1 and 4.2.1).

5.1.3. A C^r-atlas (or simply atlas) of a set X is a set of pairwise C^r-compatible charts of X whose domains have X itself as their union. Two atlases A and B of X are said to be C^r-equivalent if A ∪ B is an atlas. The relation of C^r-equivalence between atlases is an equivalence relation.

5.1.4. Let S be a set of Banach spaces. We say that an atlas A is of type S if E ∈ S for every chart c = (U, φ, E) of A. In an analogous manner, we say that an atlas A is of Hilbert type (resp. of countable-dimensional Hilbert type) if E is a Hilbert space (resp. a Hilbert space of countable type) for every chart (U, φ, E) of A.

5.1.5. A K-manifold of class C^r (or a manifold of class C^r over K, or simply a manifold when there can be no ambiguity about K and r) is a set X endowed with a class of equivalent atlases (Ens., chap. II, § 6, No. 9) for the relation of C^r-equivalence. An atlas of this class is called an atlas of the manifold X. A chart belonging to an atlas of X is called a chart of the manifold X. A chart of X whose domain contains a point a ∈ X is called a chart of X at a. A chart centred at a is a chart (U, φ, E) of X at a such that φ(a) = 0.

If X is a set and A an atlas of X, the set X endowed with the equivalence class of A is called the manifold defined by A.

When r ≠ ω (hence K = R), a manifold of class C^r is sometimes called a differential manifold. A manifold of class C^ω is also called an analytic manifold over K (or K-analytic manifold).

When in addition $K = \mathbf{R}$ (resp. $\mathbf{C}$, $\mathbf{Q}_p$), we also say real analytic manifold (resp. complex analytic manifold, $p$-adic analytic manifold).

5.1.6. Let $X$ be a manifold. The subsets of $X$ which are unions of domains of charts of $X$ form the set of open subsets for a topology on $X$, called the topology underlying the manifold structure of $X$. For every chart $c = (U, \varphi, E)$ of $X$, the mapping $\varphi$ is a homeomorphism of the open subset $U$ endowed with the topology induced by that of $X$ onto the open subset $\varphi(U)$ of $E$.

The topological space underlying $X$ is a Baire space. When $K$ is equal to $\mathbf{R}$ or $\mathbf{C}$, it is locally connected.

Let $X$ be a manifold and let $\mathscr{A}$ be an atlas of $X$. For the topological space $X$ to be separated, it is necessary and sufficient that the following condition be satisfied: whatever the charts $(U, \varphi, E)$ and $(V, \psi, F)$ belonging to the atlas $\mathscr{A}$, the graph of the mapping $\psi \circ \varphi^{-1}$ of $\varphi(U \cap V)$ into $\psi(U \cap V)$ is closed in $\varphi(U) \times \psi(V)$.

Let $X$ be a manifold; suppose that the topological space $X$ is regular. Then for every point $a \in X$, there exists a chart $(U, \varphi, E)$ of $X$ at $a$ having the following property: for a subset $Y$ of $U$ to be closed in $X$, it is necessary and sufficient that its image $\varphi(Y)$ be closed in $E$. If the space $X$ is paracompact, there exists on $X$ a distance compatible with the topology of $X$ and making $X$ a complete metric space.

5.1.7. Let $\mathfrak{S}$ be a set of Banach spaces. We say that a manifold is of type $\mathfrak{S}$ (resp. of Hilbert type, resp. of countable-dimensional Hilbert type) if it possesses an atlas of type $\mathfrak{S}$ (resp. of Hilbert type, resp. of countable-dimensional Hilbert type). If $\mathfrak{S}$ is reduced to a single element $E$, a manifold of type $\mathfrak{S}$ is also called a purely manifold of type $E$.

A pure manifold of dimension $n$ is a pure manifold of type $K^n$. We say that a manifold is locally of finite dimension if it is of type $\mathfrak{S}$ with $\mathfrak{S} = \{ K^n ; n \in \mathbf{N} \}$.

5.1.8. Let $X$ be a manifold and let $x \in X$. The dimension (finite or $+\infty$) of a chart of $X$ at $x$ depends only on $x$. It is called the dimension of $X$ at $x$ and is denoted by $\dim_x X$. The dimension of $X$, denoted by $\dim X$, is the supremum of the $\dim_x X$ for $x \in X$.

The mapping $x \mapsto \dim_x X$ is locally constant. The manifold $X$ is locally of finite dimension (resp. purely of dimension $n$) if and only if $\dim_x X < +\infty$ (resp. $\dim_x X = n$) for every $x \in X$.

5.1.9. Suppose $K$ is locally compact. Let $X$ be a separated manifold. Then, for $X$ to be locally of finite dimension, it is necessary and sufficient that $X$ be locally compact.

No. 5.2

5.1.10. Let X be a manifold and let $\xi^1, \ldots, \xi^n$ be mappings of a subset U of X into K. We say that $\xi = (\xi^1, \ldots, \xi^n)$ is a system of coordinates of X in U if the triplet $(U, \xi, K^n)$ is a chart of X; this chart is also denoted by $(U; \xi)$ or $(U; \xi^1, \ldots, \xi^n)$. If $a \in U$, we also say that $\xi$ is a system of coordinates of X at $a$; if in addition $\xi^i(a) = 0$ for $i = 1, 2, \ldots, n$, we say that the system of coordinates $\xi$ is centred at $a$.

5.2. Examples of manifolds

5.2.1. Let X be a set; there exists on X one and only one manifold structure for which the underlying topological space is discrete; this structure is a purely 0-dimensional manifold structure.

5.2.2. Let E be a Banach space. The triplet $c = (E, \mathrm{Id}_E, E)$ is a chart of E and $\mathscr{A} = \{c\}$ is an atlas of E, and hence defines a purely manifold structure of type E on E; the underlying topology is the topology given on E. Whenever in what follows we speak of the manifold structure on E, it will always be the preceding structure to which we refer.

In particular, this applies to every finite-dimensional vector space over K, endowed with the unique separated topology compatible with its vector structure (Esp. Vect. Top., chap. I, § 2, No. 3).

5.2.3. Let X be a manifold and U an open subset of X. There exists on U a manifold structure whose charts are the charts of the manifold X with domain contained in U. This structure is said to be induced by that of X (cf. No. 5.3.4); endowed with this structure, U is called an open submanifold of X.

In particular, every open subset of a Banach space E is endowed with a canonical purely manifold structure of type E. Let X be a manifold; for a triplet $(U, \varphi, E)$ to be a chart of X, it is necessary and sufficient that U be open in X and that $\varphi$ be an isomorphism of the open submanifold U of X onto an open submanifold of E.

5.2.4. Let X be a set, and $(X_i)_{i \in I}$ a covering of X. Suppose that each $X_i$ is endowed with a manifold structure such that the following condition is fulfilled:

For any $i$ and $j$ in I, the set $X_i \cap X_j$ is open in $X_i$ and $X_j$ and the manifold structures induced on $X_i \cap X_j$ by those of $X_i$ and $X_j$ coincide.

There then exists on X one and only one manifold structure such that $X_i$ is an open submanifold of X for every $i$ in I. We say that the manifold X is obtained by gluing the manifolds $X_i$.

5.2.5. Let X be a manifold. The set $X_n$ of points $x$ of X such that dim_x X = n (n integer $\geq 0$) is an open submanifold of X, which is purely of dimension n.

5.2.6. Let E be a Banach space. The set G(E) of vector subspaces of E admitting a topological supplement can be endowed with an analytic manifold structure in the following way: for every pair (F_0, G_0) $\in G(E) \times G(E)$ such that E = F_0 $\oplus$ G_0, let U_{G_0} denote the set of F $\in G(E)$ admitting G_0 as a supplement, and define a bijection $\varphi_{F_0, G_0}$ of U_{G_0} onto the Banach space $\mathscr{L}(F_0; G_0)$ by associating to each F $\in U_{G_0}$ the mapping from F_0 into G_0 having as graph the subspace F of E = F_0 $\times$ G_0. The charts (U_{G_0}, $\varphi_{F_0, G_0}$, $\mathscr{L}(F_0; G_0)$) form an atlas of G(E). Endowed with the manifold structure defined by this atlas, G(E) is called the Grassmann manifold of E.

The topological space G(E) is metrisable. If K is locally compact and E finite-dimensional, G(E) is compact.

For every integer $n \geq 0$, the set G_n(E) (resp. G^n(E)) of F $\in G(E)$ of dimension (resp. codimension) n is open and closed in G(E) and is a purely open submanifold of G(E). If K = R or C, G_n(E) is connected for every n.

The mapping which to F $\in G(E)$ associates its orthogonal in the dual E’ of E is a morphism of G(E) into G(E’) which induces an isomorphism of G^n(E) onto G_n(E’) for every integer n.

If K = R or C, or if E is finite-dimensional, G_1(E) is nothing other than the projective space deduced from E, which is thus endowed with an analytic manifold structure.

5.3. Functions of class $C^r$ and morphisms of manifolds

5.3.1. Let X be a manifold of class C', F a separated polynormed space and f a mapping from X into F. One says that f is of class C' if for every chart (V, $\varphi$, E) of X, the mapping $f \circ \varphi^{-1}$ of $\varphi(V)$ into F is of class C'. For this, it is enough that this condition be satisfied for the charts of an atlas of X. The set of mappings of class C' from X into F forms a vector subspace of the space of all mappings from X into F. It is denoted by $\mathscr{C}'(X; F)$.

When F = K, one sets $\mathscr{C}'(X; K) = \mathscr{C}'(X)$; this is a sub-K-algebra of the algebra of mappings from X into K. The elements of $\mathscr{C}'(X)$ are also called morphic functions.

When X is an open subset of a Banach space, this terminology agrees with that of Nos. 2.3.1, 3.2.1 and 4.2.1.

5.3.2. Let X and Y be two manifolds of class C' and let f be a mapping from X into Y. One says that f is of class C' or is a morphism of manifolds (of class C') if it is continuous and if, for every chart (V, $\psi$, F) of Y, the mapping $\psi \circ f$ of the open submanifold $f^{-1}(V)$ into the Banach space F is of class $C'$. For this, it is enough that there exist an atlas $\mathscr{A}$ of Y such that, for every chart $(V, \psi, F) \in \mathscr{A}$, the set $f^{-1}(V)$ be open in X and the mapping $\psi \circ f$ of the open submanifold $f^{-1}(V)$ of X into F be of class $C'$. The set of morphisms from X into Y is denoted by $\mathscr{C}'(X; Y)$. When Y is a Banach space endowed with its canonical manifold structure, the definitions of 5.3.1. and 5.3.2 are coherent. A mapping of class $C^\omega$ is also called a K-analytic mapping (or simply analytic). When $K = \mathbf{C}$, one also says holomorphic mapping.

Let $(U, \varphi, E)$ be a chart of X and $(V, \psi, F)$ a chart of Y such that $f(U) \subset V$. The mapping $\psi \circ f \circ \varphi^{-1}$ of the open set $\varphi(U)$ of E into the open set $\psi(V)$ of F is called the expression of f in the given charts.

5.3.3. Suppose X and Y finite-dimensional; let $a \in X,\ b \in Y$ and $f$ a mapping from X into Y with $f(a) = b$. Suppose first that $f$ is of class $C'$ and consider coordinate systems $(U; \xi^1, \ldots, \xi^m)$ of X at $a$ and $(V; \eta^1, \ldots, \eta^n)$ of Y at $b$ respectively, with $f(U) \subset V$. Let $\xi$ be the mapping $(\xi^1, \ldots, \xi^m)$ of U into $K^m$. Then there exist functions $u^j$ of class $C'$ on the open set $\xi(U)$ of $K^m$, with values in K, such that the coordinates of a point $y = f(x)$ of Y, (for x in U), are given by the formulae:

(1)

(2)

faisceau de fonctions à valeurs dans $Y$ if it satisfies the following condition:

Let $(U_i)_{i \in I}$ be a family of open sets of $X$, with union $U$, and let $f$ be a mapping of $U$ into $Y$; in order that $f$ belong to $\mathscr{L}(U)$, it is necessary and sufficient that $f|U_i$ belong to $\mathscr{L}(U_i)$ for every $i$ in $I$.

5.4.2. Let $X$ and $Y$ be two manifolds; for every open set $U \subset X$, let $\mathscr{L}(U)$ be the set of morphisms of $U$ into $Y$; then $\mathscr{L}$ is a sheaf of mappings with values in $Y$.

When $Y = K$, the sheaf thus defined is denoted by $\mathscr{C}_X^{r'}$.

5.4.3. Let $X$ be a topological space and let $\mathscr{S}$ be a set of Banach spaces. For every $E \in \mathscr{S}$, let $\mathscr{F}_E$ be a sheaf of mappings on $X$ with values in $E$. Suppose that the family of the $\mathscr{F}_E$, for $E \in \mathscr{S}$, satisfies the following condition:

For every $x \in X$, there exist an open neighbourhood $U$ of $x$, a space

$E_0 \in \mathscr{S}$, and a homeomorphism $\varphi$ of U onto an open set of E_0, such that, for every open set $V \subset U$ and every $E \in \mathscr{S}$, the set $\mathscr{F}_E(V)$ consists of the functions $g \circ \varphi$, where $g$ ranges over the space $\mathscr{C}^r(\varphi(V); E)$.

Then there exists on X a structure of manifold of class $C^r$ and of type $\mathscr{S}$, and only one, compatible with the given topology on X and for which $\mathscr{F}_E$ is the sheaf of functions of class $C^r$ on X with values in E.

5.4.4. Let X be a topological space and let $\mathscr{F}$ be a sheaf of functions with values in K, satisfying the following condition: for every point x of X, there exist an integer n, an open neighbourhood U of x and a homeomorphism $\varphi$ of U onto an open set of $K^n$, such that, for every open set V of U, the set $\mathscr{F}(V)$ consists of the functions $g \circ \varphi$ where $g$ ranges over the set of functions of class $C^r$ with values in K on the open set $\varphi(V)$ of $K^n$.

Then there exists on X a structure of locally finite-dimensional manifold of class $C^r$ and only one such that $\mathscr{F} = \mathscr{C}^r_X$.

5.4.5. Let $X$ and $X'$ be two $C^r$-class varieties, locally of finite dimension, and let $f$ be a continuous mapping of X into X'. For $f$ to be a morphism, it is necessary and sufficient that for every open set U' of X' and every function $g \in \mathscr{C}^r(U'; K)$, the function $g \circ f$ belong to $\mathscr{C}^r(U; K)$, where $U = f^{-1}(U')$.

5.5. Tangent spaces, tangent linear mappings

5.5.1. Let X be a variety and let $a \in X$. Consider the pairs $(c, h)$ where $c = (U, \varphi, E)$ is a chart of the variety X at $a$ and where $h$ is an element of E. Two such pairs $(c, h)$ and $(c', h')$ are said to be equivalent if the derivative at $\varphi(a)$ of the mapping $\varphi' \circ \varphi^{-1}$ (which is defined on a neighbourhood of $\varphi(a)$) transforms $h$ into $h'$. We thus obtain an equivalence relation between pairs $(c, h)$ and call a tangent vector at $a$ to X a class of equivalent pairs $(c, h)$ (Ens., Chapter II, § 6, No. 9).

The tangent vectors at $a$ to X form a set denoted by $T_a(X)$. If $c = (U, \varphi, E)$ is a chart of the variety X at $a$, the mapping $\theta_c$ of E into $T_a(X)$ which associates to an element $h$ of E the tangent vector represented by the pair $(c, h)$ is a bijection. If one transports to $T_a(X)$, by means of $\theta_c$, the structure of topological K-vector space of E, the structure thus obtained does not depend on the choice of $c$ and makes $T_a(X)$ a Banach space, called the tangent space to X at $a$. The dimension (finite or $+\infty$) of $T_a(X)$ is equal to the dimension of X at $a$.

5.5.2. Let X, Y be two manifolds, $f$ a morphism of X into Y and $a$ a point of X. Consider a chart $c = (U, \varphi, E)$ of X at $a$ and a chart $c' = (V, \psi, F)$ of Y at $b$ with $f(U) \subset V$; the mapping $\Phi = \psi \circ f \circ \varphi^{-1}$ of $\varphi(U)$ into F is of class $C^r$, and its derivative $D\Phi(\varphi(a))$ at the point $\varphi(a)$ is a continuous linear mapping of E into F. The continuous linear mapping $\theta_{c'} \circ D\Phi(\varphi(a)) \circ \theta_c^{-1}$ of $T_a(X)$ into $T_b(Y)$ does not depend on the charts $c$ and $c'$ chosen; it is denoted by $T_a(f)$ and is called the tangent linear mapping to $f$ at $a$. If $g$ is a morphism of $Y$ into a manifold $Z$, we have:

5.5.3. Let $f : X \to Y$ be a morphism of manifolds. If $f$ is locally constant, we have $T_a(f) = 0$ for every $a$ in $X$. Conversely, if $T_a(f) = 0$ for every $a \in X$ and if the field $K$ is of characteristic 0, then $f$ is locally constant.

5.5.4. Let $U$ be an open set of a manifold $X$ and $f$ the canonical injection of $U$ into $X$; for every point $a$ of $U$, the mapping $T_a(f)$ of $T_a(U)$ into $T_a(X)$ is an isomorphism of topological vector spaces, by means of which these two spaces will henceforth be identified.

5.5.5. Let $U$ be an open set of a Banach space $E$; if $\iota$ is the canonical injection of $U$ into $E$, the triple $c = (U, \iota, E)$ is a chart of $U$, and the atlas $\{c\}$ defines the manifold structure of $U$. Given a point $a$ of $U$, the mapping $\theta_c$ is an isomorphism of topological vector spaces of $E$ onto $T_a(U)$; the inverse isomorphism will be denoted by $\zeta_E(a)$.

Let $f$ be a mapping of class $C^r$ of $U$ into an open set $V$ of a Banach space $F$; for every point $a$ of $U$, we have:

where $Df(a)$ is the derivative of $f$ at $a$ (1.2.1). In other words, the diagram:

is commutative.

5.5.6. A cotangent vector to $X$ at $a$, or also a tangent covector or simply a covector at $a$, is any continuous linear form on the topological vector space $T_a(X)$; these covectors are therefore the elements of the topological dual $T_a(X)'$ of the tangent space to $X$ at $a$. This space will be endowed with the topology of uniform convergence on the bounded subsets of $T_a(X)$: endowed with this topology, $T_a(X)'$ is a Banach space which is called the cotangent space to $X$ at $a$. It is also denoted by $T'_a(X)$.

Let $f$ be a morphism of $X$ into a Banach space $E$. The differential of $f$ at $a$, denoted by $d_a f$ or $df_a$, is the continuous linear mapping $\zeta_E(f(a)) \circ T_a(f)$ of $T_a(X)$ into $E$. For $t \in T_a(X)$, one sometimes denotes by t(f) or t . f the value $d_a f(t)$ of the linear mapping $d_a f$ at the point $t$. The mapping $f \mapsto d_a f$ is linear.

If $E = K$, the differential $d_a f$ of $f$ at $a$ is a covector of $X$ at $a$.

Let $E, F, G$ be three Banach spaces and let $(u, v) \mapsto u . v$ be a continuous bilinear mapping of $F \times G$ into $E$. Let $f$ (resp. $g$) be a morphism of $X$ into $F$ (resp. $G$). Then the mapping $f . g : x \mapsto f(x) . g(x)$ is a morphism of $X$ into $E$ and, for every $t \in T_a(X)$, we have:

Take in particular $G = E$ and $F = K$, the bilinear mapping considered being multiplication. We then have:

5.5.7. Let $X$ be a locally finite-dimensional variety and let $\xi^1, \ldots, \xi^m$ be morphic functions defined in an open neighbourhood $U$ of a point $a$ of $X$. The following conditions are equivalent:

(i) there exists an open neighbourhood $V$ of $a$ contained in $U$, such that the functions $\xi^i|V$ (for $1 \leq i \leq m$) form a system of coordinates of $X$ in $V$;

(ii) the differentials $d_a \xi^i$ for $1 \leq i \leq m$ form a basis of $T_a(X)'$;

(iii) the mapping $\xi = (\xi^1, \ldots, \xi^m)$ of $U$ into $K^m$ is étale at $a$ (see No. 5.7.6).

For the differentials $d_a \xi^1, \ldots, d_a \xi^m$ to be linearly independent, it is necessary and sufficient that there exist a neighbourhood $V$ of $a$, contained in $U$, such that the functions $\xi^i|V$ form part of a system of coordinates of $X$ in $V$.

5.5.8. Let $X$ be a locally finite-dimensional variety and let $\xi = (\xi^1, \ldots, \xi^m)$ be a system of coordinates of $X$ in an open set $U$. Let $a \in U$: we denote by $(\partial_{1,a}, \ldots, \partial_{m,a})$ the basis of the tangent space $T_a(X)$ dual to the basis $(d_a \xi^1, \ldots, d_a \xi^m)$ of $T_a(X)'$. We therefore have:

The tangent vector $\partial_{i,a}$ is also denoted by $(\partial/\partial \xi^i)_a$.

Let $f$ be a function of class $C^r$ on $U$, with values in a Banach space $E$. We denote by $\partial_i f$ or $\partial f/\partial \xi^i$ the function $a \mapsto \partial_{i,a}(f)$: this is a function of class $C^{r-1}$ on $U$ with values in $E$ (a continuous function if $r = 1$). Its value at a point $a$ of $U$ is sometimes denoted by $(\partial f/\partial \xi^i)_a$. For all $t \in T_a(X)$, we have:

which may also be written:

These notations are consistent with those of 1.6.3.

5.5.9. Let $X$ and $Y$ be two varieties and $f$ a morphism of $X$ into $Y$. The rank of $f$ at a point $a$ of $X$, denoted $\mathrm{rg}_a f$, is the rank (finite or equal to $+\infty$) of the linear mapping $T_a(f)$. The mapping $a \mapsto \mathrm{rg}_a f$ is lower semicontinuous.

5.6. Products of varieties

5.6.1. Let $X$ and $X'$ be two sets and let $c = (U, \varphi, E)$ (resp. $c' = (U', \varphi, E')$) be a chart of $X$ (resp. $X'$). The triple

is a chart of the product set $X \times X'$, denoted $c \times c'$.

5.6.2. Let $X$ and $X'$ be two varieties of class $C^r$. There exists on the product set $X \times X'$ one and only one structure of a variety of class $C^r$ such that $c \times c'$ is a chart of $X \times X'$ for every chart $c$ of $X$ and every chart $c'$ of $X'$. Endowed with this structure, $X \times X'$ is called the product variety of the varieties $X$ and $X'$. One defines analogously the product of any finite number of varieties. The topology of the variety $X \times X'$ is the product topology of the topologies of $X$ and $X'$. For $a \in X$ and $b \in X'$, one has:

5.6.3. Let $X$ and $X'$ be two varieties, and let $X \times X'$ be their product. Let $a \in X$ and let $a' \in X'$. The canonical projections

are morphisms. Let $\pi_i \ (i = 1, 2)$ be their tangent linear mappings at the point $(a, a')$. The mapping

is an isomorphism, which makes it possible to identify the tangent space to $X \times X'$ at $(a, a')$ with the product $T_a(X) \times T_{a'}(X')$.

The injection of $T_a(X)$ into $T_{(a,a')}(X \times X')$ resulting from this identification is the tangent linear mapping at $a$ of the morphism $x \mapsto (x, a')$ of $X$ into $X \times X'$; there is an analogous result for the injection of $T_{a'}(X')$ into $T_{(a,a')}(X \times X')$.

5.6.4. Let $W, X$ and $X'$ be three varieties, and let $f : W \to X, f' : W \to X'$ be two mappings. For the mapping

to be a morphism, it is necessary and sufficient that $f$ and $f'$ be morphisms (this justifies the use of the term « product », cf. Ens., Chap. IV, § 2, No. 4). In this case, if $a$ is a point of $W$, we have:

taking into account the identification made above.

5.6.5. Let $f : X \to Y$ and $f' : X' \to Y'$ be morphisms of varieties. Then $f \times f' : X \times X' \to Y \times Y'$ is a morphism. If $a \in X$ and $a' \in X'$, we have:

5.6.6. Let $X_1, X_2$ and $Z$ be three varieties and $f$ a morphism of $X_1 \times X_2$ into $Z$. Let $a \in X_1$ and $b \in X_2$. The tangent linear mapping to the partial mapping $x \mapsto f(x, b)$ (resp. $y \mapsto f(a, y)$) of $X_1$ (resp. $X_2$) into $Z$ is denoted by $T^1_{(a, b)}(f)$ (resp. $T^2_{(a, b)}(f)$). If we identify $T_{(a, b)}(X_1 \times X_2)$ with $T_a(X_1) \times T_b(X_2)$, we have:

for every $u \in T_a(X_1)$ and every $v \in T_b(X_2)$.

5.6.7. (« Theorem on implicit functions »). With the hypotheses and notation of the preceding number, suppose in addition that $T^2_{(a, b)}(f)$ is bijective. There then exists an open neighbourhood $U$ of $a$ in $X_1$ and an open neighbourhood $V$ of $b$ in $X_2$ having the following property: for every $x \in U$, there exists one and only one point $g(x)$ of $V$ such that $f(x, g(x)) = f(a, b)$, and the mapping $g$ is a morphism of $U$ into $V$. For every $x \in U$, we have:

5.7. Immersions, étale morphisms

5.7.1. Let $X$ and $Y$ be two manifolds, $f$ a morphism of $X$ into $Y$ and $a$ a point of $X$. Put $b = f(a)$. The following conditions are equivalent: (i) The linear mapping $T_a(f)$ is injective and its image is a closed vector subspace of $T_b(Y)$ admitting a topological supplement $^1$;

(ii) There exist a Banach space $F$, a closed vector subspace $E$ of $F$ admitting a topological supplement, and charts $(U, \varphi, E)$ of $X$ at $a$ and $(V, \psi, F)$ of $Y$ at $b$ such that $f(U) \subset V$ and $\varphi = \psi \circ (f|U)$.

(iii) There exist an open neighbourhood $U$ of $a$, an open neighbourhood $V$ of $b$ containing $f(U)$, a manifold $Z$, a point $c$ of $Z$, and an isomorphism of manifolds $g$ of $U \times Z$ onto $V$ such that $f(x) = g(x, c)$ for all $x \in U$.

(iv) There exist an open neighbourhood $U$ of $a$, an open neighbourhood $V$ of $b$ and a morphism $q$ of $V$ into $X$ with $f(U) \subset V$, and $q(f(x)) = x$ for all $x \in U$.

When $X$ and $Y$ are finite-dimensional, the preceding conditions are also equivalent to the following:

(v) There exist an open neighbourhood $U$ of $a$, a coordinate system $(\eta^1, \ldots, \eta^n)$ of $Y$ in an open neighbourhood $V$ of $b$ containing $f(U)$ and an integer $m \leq n$ such that $\eta^j \circ f = 0$ for $m < j \leq n$ and the functions $\eta^1 \circ f, \ldots, \eta^m \circ f$ form a coordinate system of $X$ in $U$.

If properties (i) to (iv) are satisfied, we say that $f$ is an immersion at $a$.

The set of points where $f$ is an immersion is open in $X$; if this open set is all of $X$, we say that $f$ is an immersion.

For $f$ to be an immersion, it is necessary and sufficient that $X$ can be covered by open sets $U_i$ such that, for every $i, f|U_i$ is an isomorphism of $U_i$ onto a subvariety of $Y$ (cf. No. 5.8.3).

5.7.2. Examples: (a) If $X$ is an open set in a variety $Y$, the canonical injection of $X$ into $Y$ is an immersion. (b) Let $E$ and $F$ be two Banach spaces and let $u$ be a continuous linear mapping of $E$ into $F$. Then $u$ is an immersion if and only if $u$ is an isomorphism of $E$ onto a closed vector subspace of $F$ admitting a topological supplement.

5.7.3. Let $f : X \to Y$ and $g : Y \to Z$ be two morphisms. If $f$ and $g$ are immersions, $g \circ f$ is an immersion. Conversely, if $g \circ f$ is an immersion, then $f$ is an immersion. If $f : X \to Y$ and $f' : X' \to Y'$ are immersions, $f \times f'$ is an immersion.

5.7.4 Suppose that $X$ and $Y$ are varieties of finite dimension over a field $K$ of characteristic 0. If $f : X \to Y$ is an injective morphism, the set of points of $X$ where $f$ is an immersion is an open dense set

$^1$ Every closed vector subspace of finite codimension of a Banach space admits a topological supplement; if $K = \mathbf{R}$ or $\mathbf{C}$, the same is true of every finite-dimensional vector subspace (necessarily closed).

in $X$. If K = R or C, this result remains true if one supposes only that X is of finite dimension.

5.7.5. Let $f : X \to Y$ be an immersion and let $g$ be a continuous mapping of a variety Z into X. For $g$ to be a morphism, it is necessary and sufficient that $f \circ g$ be a morphism.

5.7.6. Let $f : X \to Y$ be a morphism, and let $a$ be in X. The following two properties are equivalent: (i) $T_a(f)$ is bijective. (ii) There exists an open neighbourhood U of $a$ and an open neighbourhood V of $f(a)$ such that $f$ induces an isomorphism of U onto V. When these properties are satisfied, one says that $f$ is a local isomorphism at $a$, or that $f$ is étale at $a$. If this holds for every $a \in X$, one says that $f$ is étale, or is an étalement, or that X is étalé in Y (by means of $f$). For a morphism to be étale, it is necessary and sufficient that it be both an immersion (No. 5.7.1) and a submersion (No. 5.9.1.).

5.7.7. Let $f : X \to Y$ be an immersion; suppose the variety X is purely finite-dimensional. Then $f$ is étale in each of the following two cases: (i) $\dim Y = \dim X$; (ii) $f$ is surjective and the topology of X has a countable basis of open sets.