§ 12. Jets

In this paragraph, we denote by $X$ and $Y$ two varieties of class $C^r$, where $r \in \mathbf{N}_k$, and $k$ an integer such that $0 \leq k \leq r$.

From No. 12.3 onwards, we assume:

— either that $K$ is of characteristic zero;

— or that the varieties, Banach spaces and vector bundles considered are locally of finite dimension.

12.1. Jets of mappings

12.1.1. Let $x \in X$ and let $f, g$ be two continuous mappings defined in a neighbourhood of $x$ and taking values in $Y$. We say that $f$ and $g$ have contact of order $\geq k$ at $x$ if $f(x)=g(x)$ and if there exist charts $(U,\varphi,E)$ of $X$ at $x$ and $(V,\psi,F)$ of $Y$ at $f(x)$ such that the mappings $\psi \circ f \circ \varphi^{-1}$ and $\psi \circ g \circ \varphi^{-1}$, defined in a neighbourhood of $\varphi(x)$ in $E$ and taking values in $F$, have contact of order $\geq k$ at $\varphi(x)$ (1.1.2). This property is then satisfied by all charts of $X$ at $x$ and of $Y$ at $f(x)$.

12.1.2. Let $x \in X$, $y \in Y$. In the set of mappings of class $C^r$ defined in a neighbourhood of $x$, taking values in $Y$, and mapping $x$ onto $y$, the relation « $f$ and $g$ have contact of order $\geq k$ » is an equivalence relation; the class of $f$ for this relation is denoted by $j^k_x(f)$ and is called the jet of order $k$ of $f$ with source $x$ and target $y$. The source of a jet is denoted by $s(j)$ and its target by $b(j)$.

The set of jets of order $k$ from $X$ into $Y$ (resp. with source $x$, resp. with target $y$) is denoted by $J^k(X,Y)$ (resp. $J^k_x(X,Y)$, resp. $J^k(X,Y)_y$) and we put

12.1.3. If $U$ and $V$ are open subsets of $X$ and $Y$ respectively, one identifies in an obvious way $J^k(U,V)$ with the inverse image of $U \times V$ by the mapping

12.1.4. Let $Z$ be a variety of class $C^r$, and let $(x,y,z) \in X \times Y \times Z$. Let $j \in J^k(X,Y)_y$ and let $j' \in J^k(Y,Z)_z$. The mappings $f' \circ f$, with $f \in j$ and $f' \in j'$, have the same jet of order $k$ at $x$; this jet is called the composite of $j$ and $j'$ and is denoted by $j' \circ j$; one has $s(j' \circ j)=s(j)$ and $b(j' \circ j)=b(j')$. If $T$ is a variety of class $C^r$ and if $j'' \in J^k_z(Z,T)$, one has

12.1.5. Let $j \in J_x^k(X, Y)$, with $x \in X$, and let $k'$ be an integer such that $0 \leq k' \leq k$. The jets $j_x^{k'}(f)$, for $f \in j$, are equal; the jet thus defined is denoted by $r^{k', k}(j)$; the mapping $r^{k, k'} : J^k(X, Y) \to J^{k'}(X, Y)$ is surjective.

12.1.6. Suppose that $r$ is equal to $\infty$ or to $\omega$. Let $f$ and $g$ be two mappings of class $C^r$ defined in a neighbourhood of a point $x \in X$, and with values in $Y$. If $j_x^m(f) = j_x^m(g)$ for every integer $m \geq 0$, one says that $f$ and $g$ have contact of infinite order at $x$. One thus obtains an equivalence relation whose classes are called the jets of infinite order at $x$; the jet of infinite order of $f$ is denoted by $j_x^\infty(f)$ or $j_x^\omega(f)$. The definitions and results which precede extend without modification to jets of infinite order.

12.2. Jets of mappings of Banach spaces

In this No., E and F denote two Banach spaces. If $m$ is an integer $\geq 0$, we denote by $P_m(E; F)$ the Banach space of continuous homogeneous polynomials of degree $m$ on E with values in F (Part 1, Appendix, A.2).

12.2.1. Let U be an open subset of E, let $f : U \to F$ be a mapping of class $C^r$, and let $a \in U$. There exists one and only one continuous polynomial

12.2.2. Let U (resp. V) be an open subset of E (resp. F). Let $Q^k(E, F)$ denote the product Banach space of the $P_m(E; F)$ for $1 \leq m \leq k$. The mapping

12.3. Jet manifolds

12.3.1. Let $c = (U, \varphi, E)$ and $c' = (V, \psi, F)$ be charts of X and Y respectively. The mappings $\varphi$ and $\psi$ define, by transport of structure, a bijection $\pi$ of $J^k(U, V)$ onto

If $(x, y) \in X \times Y$, the sets $J_x^k(X, Y), J^k(X, Y)_y$ and $J_x^k(X, Y)_y$ are closed submanifolds of $J^k(X, Y)$.

12.3.2. If $X$ and $Y$ are pure (resp. finite-dimensional, resp. separated, resp. connected), the same is true of $J^k(X, Y)$.

If $U$ (resp. $V$) is open in $X$ (resp. in $Y$), $J^k(U, V)$ is an open submanifold of $J^k(X, Y)$, cf. 12.1.3.

12.3.3. The mappings $s : J^k(X, Y) \to X, b : J^k(X, Y) \to Y$ and $(s, b) : J^k(X, Y) \to X \times Y$ are fibrations (6.1.1) of class $C^{r-k}$. When $k = 0$, $(s, b)$ is an isomorphism.

12.3.4. Let $T_X$ and $T_Y$ denote the vector bundles $pr_1^* T(X)$ and $pr_2^* T(Y)$ on $X \times Y$, and let $\mathscr{L}(T_X; T_Y)$ be the bundle of homomorphisms from $T_X$ into $T_Y$ (7.7.3); if $(x, y) \in X \times Y$, we have

For $X = K$, this isomorphism identifies $J_0^1(K, Y)$ with $T(Y)$; for $Y = K$, it identifies $J^1(X, K)_0$ with the dual $T'(X)$ of $T(X)$.

12.3.5. Let $k'$ be an integer such that $0 \leqslant k' \leqslant k$. The mapping

12.3.6. Let $Z$ be a variety of class $C^r$, and let $T$ be the set of pairs

12.3.7. If $f : X \to Y$ is a morphism of class $C^r$, the mapping $x \mapsto j_x^k(f)$ is a morphism $j^k(f)$ of class $C^{r-k}$ from $X$ into $J^k(X, Y)$.

12.3.8. Let $k'$ and $k''$ be positive integers whose sum is $k$. Let $x \in X$, let $U$ be an open neighbourhood of $x$, and let $f : U \to Y$ be a morphism of class $C^r$. The mapping

12.3.9. Let $X', Y'$ be manifolds of class $C^r$, let $f : X \to X'$ and $g : Y' \to Y$ be morphisms, and let $(x, y') \in X \times Y'$, $x' = f(x)$, $y = g(y')$. If $u \in J_{x'}^k(X', Y')_{y'}$, put

12.3.10. Let $A$ be a compact subset of $X$. Let $\mathscr{C}^k(X; Y)$ be the set of mappings of class $C^k$ from $X$ into $Y$. For every $f \in \mathscr{C}^k(X; Y)$, the mapping $j^k(f)|A$ belongs to the set $\mathscr{C}(A; J^k(X, Y))$ of continuous mappings from $A$ into $J^k(X, Y)$. We have thus defined a mapping $\lambda$ from $\mathscr{C}^k(X; Y)$ into $\mathscr{C}(A; J^k(X, Y))$. The inverse image by $\lambda$ of the topology of compact convergence on $\mathscr{C}(A; J^k(X, Y))$ (TG, X, § 3, Def. 1) is a topology on $\mathscr{C}^k(X; Y)$; it is called the topology of uniform $C^k$-convergence on $A$.

12.4. Frames and principal fibrations

12.4.1. Let $j \in J^k(X, Y)$ and let $x = s(j)$, $y = b(j)$. We say that $j$ is invertible if there exists $j' \in J_y^k(Y, X)_x$ such that $j' \circ j = j_x^k(\mathrm{Id}_X)$ and $j \circ j' = j_y^k(\mathrm{Id}_Y)$; the jet $j'$ is then determined uniquely; we denote it by $j^{-1}$. If $k = 0$, every jet is invertible. If $k \geqslant 1$, the following conditions are equivalent: a) $j$ is invertible; b) the mapping $T(j) : T_x(X) \to T_y(Y)$ (cf. 12.3.4) is an isomorphism; c) there exists an isomorphism $g$ of class $C^r$ of an open neighbourhood of $x$ onto an open neighbourhood of $y$, such that $j_x^k(g) = j$.

12.4.2. Let $E$ be a Banach space. We denote by $\mathbf{GL}^k(E)$ the set of jets of order $k$ from $E$ into $E$ which are invertible and have $0$ as source and target; it is an open set of $J_0^k(E, E)_0$. Endowed with the law of composition of jets, and with the structure of variety induced by that of the Banach space $J_0^k(E, E)_0 = Q^k(E, E)$, it is a group variety of class $C^\omega$.

We have $\mathbf{GL}^0(E) = \{ e \}$. The group $\mathbf{GL}^1(E)$ is identified, by means of $T$, with the group $\mathbf{GL}(E)$ of automorphisms of $E$.

If $k' \leqslant k$, the mapping $r^{k, k'} : \mathbf{GL}^k(E) \to \mathbf{GL}^{k'}(E)$ is a homomorphism of class $C^\omega$ and is a surjective submersion. The mapping $f \mapsto \mathrm{Id}_E + f$ is an isomorphism of group varieties from $P_k(E, E)$ onto the kernel of $r^{k, k-1}$.

12.4.3. Let E be a Banach space. For every $x \in X$, an E-frame of order k of X at x is called any invertible element of $J_0^k(E, X)_x$. The set of E-frames of order k of X is an open subvariety $R^k(E, X)$ of $J_0^k(E, X)$; the mapping b has as its restriction a morphism, still denoted by b, from $R^k(E, X)$ into X; analogously, if $k' \leq k$, we still denote by $r^{k, k'}$ the morphism from $R^k(E, X)$ into $R^{k'}(E, X)$, restriction of the mapping $r^{k, k'}$ of 12.1.5.

12.4.4. We suppose that X is purely of type E (5.1.7). The group $GL^k(E)$ operates on the right on $R^k(E, X)$ by the law $(\rho, u) \mapsto \rho \circ u$ and the quadruplet $\lambda_X = (R^k(E, X), GL^k(E), X, b)$ is a principal fibration (6.2.1) with structural group $GL^k(E)$, basis X and of class $C^{r-k}$.

If $k' \leq k$, let H be the kernel of $r^{k, k'} : GL^k(E) \to GL^{k'}(E)$; the quadruplet $(R^k(E, X), H, R^{k'}(E, X), r^{k, k'})$ is a principal fibration of class $C^{r-k}$.

The manifold $J^k(X, Y)$ is provided with a structure of associated fibre space to $\lambda_X$ (6.5.1): the type fibre is $J_0^k(E, Y)$ on which $GL^k(E)$ operates on the left by the law $(u, j) \mapsto j \circ u^{-1}$; the frame mapping $R^k(E, X) \times J_0^k(E, Y) \to J^k(X, Y)$ transforms $(\rho, j)$ into $j \circ \rho^{-1}$. The projection $J^k(X, Y) \to X$ corresponding to this structure of associated fibre space is s.

12.4.5. Let F be a Banach space, and suppose that Y is purely of type F. Then $J^k(X, Y)$ is provided with a structure of associated fibre space to $\lambda_Y$: the type fibre is $J^k(X, F)_0$ on which $GL^k(F)$ operates on the left by the law $(v, j) \mapsto v \circ j$; the frame mapping is $(\sigma, j) \mapsto \sigma \circ j$; the projection $J^k(X, Y) \to Y$ is b.

12.4.6. The hypotheses being those of 12.4.4 and 12.4.5, let $\mu$ be the principal fibration

12.5. Jets of sections

12.5.1. Let $\pi : Y \to X$ be a submersion, let $x \in X$, and let $s \in J_x^k(X, Y)$. We say that s is a jet of section (of order k) of $\pi$ if s is of the form $j_x^k(f)$, where f is a section of class $C^r$ of $\pi$ above an open neighbourhood of x; this condition is equivalent to

If $k' \leq k$, the mapping $r^{k, k'} : J^k(X, Y) \to J^{k'}(X, Y)$ applies $P^k(\pi)$ into $P^{k'}(\pi)$; the mapping from $P^k(\pi)$ into $P^{k'}(\pi)$ deduced from $r^{k, k'}$ is a fibration of class $C^{r-k}$.

12.5.2. Let $Z$ be a variety of class $C^r$; suppose that $Y = X \times Z$ and let $\pi = \mathrm{pr}_1 : Y \to X$. The restriction of $J^k(\mathrm{Id}_X, \mathrm{pr}_2)$ (cf. 12.3.9) to $P^k(\pi)$ is an isomorphism of $P^k(\pi)$ onto $J^k(X, Z)$, by which we identify these two varieties.

12.5.3. Let $Y'$ be a variety of class $C^r$, let $\pi : Y \to X$ and $\pi' : Y' \to X$ be submersions and let $g : Y \to Y'$ be a morphism such that $\pi' \circ g = \pi$. The mapping $J^k(\mathrm{Id}_X, g)$ has as its restriction a mapping

which is of class $C^{r-k}$.

Let $X'$ be a variety of class $C^r$, and let $f : X' \to X$ be a morphism; put $(Y', \pi') = f^*(Y, \pi)$, cf. 5.11.5. The mapping $J^k(f, \mathrm{Id}_Y)$ (cf. 12.3.9) defines a mapping of class $C^{r-k}$ from $f^*P^k(\pi)$ into $P^k(\pi')$.

12.6. Jets of sections of a vector bundle

12.6.1. Let $E$ be a vector bundle of class $C^r$ with base $X$, and let $\pi$ be its projection. Let $c_0 = (U, \varphi, F_0)$ be a vector bundle chart of $E$ and $c_1 = (U, \psi, F_1)$ be a chart of the variety $X$, with the same domain $U$. These charts define a bijection $\theta$ of $P^k(\pi)|U$ onto $J^k(\psi(U), F_0) = \psi(U) \times F_0 \times Q^k(F_1, F_0)$, cf. 12.2.2 and 12.3.1, whence a vector bundle chart

of $P^k(\pi)$. The charts so obtained form a vector bundle $C^{r-k}$-atlas, which equips $P^k(\pi)$ with the structure of a vector bundle of class $C^{r-k}$, with base $X$. This vector bundle is denoted by $P^k(E)$; the underlying variety structure is the one defined in 12.5.1.

Let $U$ be an open subset of $X$, and let $f \in \mathscr{S}_E^r(U)$ be a section of class $C^r$ of $E$ over $U$. The mapping $j^k(f) : x \mapsto j_x^k(f)$ is a section of class $C^{r-k}$ of $P^k(E)$ over $U$. The mapping $j^k : \mathscr{S}_E^r(U) \to \mathscr{S}_{P^k(E)}^{r-k}(U)$ is $K$-linear.

12.6.2. If $F$ is a Banach space, the identification (12.5.2) of $P^k(F_X)$ with $J^k(X, F)$ equips the latter variety with a structure of vector bundle of class $C^{r-k}$, with base $X$. When $F = K$, we write $P^k(X)$ instead of $P^k(K_X)$.