§ 13. Point distributions

13.1. Symmetric tensors and Banach spaces

13.1.1. Let E be a module over a commutative ring A. If n is an integer $\geq 0$, denote by $TS^n(E)$ the module of symmetric tensors of rank n of E (A, III, p. 71 and A, IV, § 5, no 2); the direct sum of the $TS^n(E)$ is denoted by $TS(E)$. If r is an integer, one of the symbols $\infty,\omega$, put

consequently, if $r\geq 0$, $TS^{r}(E)$ is the direct sum of $TS^{(0)}(E)=A$ and of $TS^{(r)+}(E)$. One has $TS^{(1)+}(E)=E$ and $TS^{(\infty)}(E)=TS^{(\omega)}(E)=TS(E)$.

If $n$ is an integer $\geq 0$, and if $x\in E$, denote by $\gamma_n(x)$ the element $x\otimes\cdots\otimes x$ of $TS^n(E)$; for $n=0$, $\gamma_0(x)=1$.

13.1.2. (« Polynomial mappings »). Suppose that A is integral infinite and that E is a free a-module. Let F be an a-module. A mapping $f:E\rightarrow F$ is said to be homogeneous polynomial of degree n if it satisfies the following equivalent conditions (cf. A, IV, § 5, no 9):

a) There exists a linear mapping $\tilde f:TS^n(E)\rightarrow F$ such that $f(x)=\tilde f(\gamma_n(x))$ for all $x\in E$.

b) There exists a multilinear mapping $u:E^n\rightarrow F$ such that $f(x)=u(x,\ldots,x)$ for all $x\in E$.

Suppose that this is the case. The mapping $\tilde f$ satisfying a) is then unique; if $t\in TS^n(E)$, denote by $\langle f,t\rangle$ the element $\tilde f(t)$. If $u:E^n\rightarrow F$ satisfies b), the corresponding linear mapping of $\otimes^nE$ into F coincides in the submodule $TS^n(E)$. When $n!$ is invertible in A, one may choose $u$ symmetric, and this in a unique way.

13.1.3. Let E be a Banach space over K and F a separated polynomial space over K. Let U be an open neighbourhood of 0 in E, $f:U\rightarrow F$ a mapping of class $C^r$ ($r\in\mathbf{N}_K$), and $t$ an element of $TS^{(r)}(E)$, cf. 13.1.1 (applied to the ring $A=K$). Let $k$ be an integer $\leq r$ such that $t\in TS^{(k)}(E)$. One can decompose $t$ and $f$ uniquely into

and

where $f_i$ is a homogeneous continuous polynomial of degree $i$ ($0 \leq i \leq k$) and where $h$ has contact of order $\geq k$ with 0 at the point 0. Put then

this is an element of F which does not depend on the choice of $k$.

13.1.4. Let E and E’ be two Banach spaces, U an open neighbourhood of 0 in E and $\varphi : U \to E'$ a mapping of class $C^r$ such that $\varphi(0) = 0$. Let $t \in TS^{(r)}(E)$. There exists one and only one element $t' \in TS^{(r)}(E')$ such that, for every separated polynormed space F, every open neighbourhood U’ of 0 in E’, and every mapping $f : U' \to F$ of class $C^r$, one has

(*)

This element $t'$ is denoted by $\varphi_*(t)$. The mapping

thus defined is linear.

13.2. Point distributions

In this No., X denotes a manifold of class $C^r$, and $x$ a point of X.

13.2.1. Let $\mathscr{A}$ be the set of pairs $(c, t)$, where $c = (U, \varphi, E)$ is a chart of X centred at $x$, and where $t \in TS^{(r)}(E)$. Two elements $((U, \varphi, E), t)$ and $((U', \varphi', E'), t')$ of $\mathscr{A}$ are said to be equivalent if $t' = \gamma_*(t)$, where $\gamma = \varphi' \circ \varphi^{-1}$. In this way one obtains an equivalence relation on $\mathscr{A}$; an equivalence class for this relation is called a point distribution at $x$ on X.

Let $T_x^{(r)}(X)$ be the set of point distributions at $x$ on X. If $c$ is a chart of X centred at $x$, the mapping

which associates with $t$ the class of $(c, t)$, is a bijection. In all that follows, $T_x^{(r)}(X)$ is endowed with the structure of a K-vector space obtained by transporting that of $TS^{(r)}(E)$ by means of $\theta_c$; this structure does not depend on the choice of $c$. Analogously, if $k \leq r$, $T_x^{(k)}(X)$ and $T_x^{(k)+}(X)$ denote the images of $TS^{(k)}(E)$ and $TS^{(k)+}(E)$ under $\theta_c$; they do not depend on the choice of $c$. An element of $T_x^{(r)}(X)$ is said to be of order $\leq k$ if it belongs to $T_x^{(k)}(X)$. One has

There exists one and only one element $\varepsilon_x$ of $T_x^{(0)}(X)$ such that $\theta_c(1) = \varepsilon_x$ for every chart $c$ of X centred at $x$. If $t$ is a point distribution at $x$, the constant term of $t$ is the element $\lambda$ of K such that $t - \lambda \varepsilon_x \in T_x^{(r)+}(X)$; one says that $t$ is without constant term if its constant term is zero.

13.2.2. Let F be a separated polynormed space, and let $f$ be a function of class $C^r$ with values in F, defined on a neighbourhood of $x$. Let $t$ be a point-supported distribution at $x$ on X.

Consider a chart $c = (U, \varphi, E)$ of $X$ centred at $x$, and put $f_c = f \circ \varphi^{-1}$ and $t_c = \theta_c^{-1}(t)$; the element $\langle f_c, t_c \rangle$ of $F$ defined in 13.1.3 does not depend on the choice of $c$; we denote it by $\langle f, t \rangle$. One has $\varepsilon_x(f) = f(x)$.¹ The constant term of $t$ is $\langle 1, t \rangle$.

If $F'$ is a separated polynormed space and if $u : F \to F'$ is continuous linear, one has $\langle u \circ f, t \rangle = u(\langle f, t \rangle)$.

Suppose that $F$ is a Banach space, and that $t$ is of order $\leq k$, with $k$ finite. Let $j \in J_x^k(X, F)$, cf. 12.1.2, and let $f \in j$. Then $\langle f, t \rangle$ depends only on the jet $j$; we denote it by $\langle j, t \rangle$. The mapping from $T_x^{(k)}(X) \times J_x^k(X, F)$ into $F$ thus defined is bilinear. When $F = K$, we have put $J_x^k(X, F) = P_x^k(X)$, cf. 12.6.2, and we obtain a bilinear form on $T_x^{(k)}(X) \times P_x^k(X)$.

13.2.3. Let $\varphi : X \to Y$ be a morphism of manifolds of class $C^r$ and let $y = \varphi(x)$. Let $c = (U, \psi, E)$ be a chart of $X$ centred at $x$ and let $c' = (U', \psi', E')$ be a chart of $Y$ centred at $y$; let $\tilde{\varphi}$ be the expression of $\varphi$ in these charts (5.3.2), and let $\tilde{\varphi}_*$ be the corresponding mapping from $TS^{(r)}(E)$ into $TS^{(r)}(E')$, cf. 13.1.4. There exists one and only one linear mapping, denoted by $T_x^{(r)}(\varphi)$ or $\varphi_*$, from $T_x^{(r)}(X)$ into $T_y^{(r)}(Y)$, making the diagram commutative

This mapping does not depend on the choice of $c$ and $c'$. If $t \in T_x^{(r)}(X)$, $\varphi_*(t)$ is called the image of $t$ by $\varphi$. We have $\varphi_*(\varepsilon_x) = \varepsilon_y$. If $k \leq r$, we have

If $\varphi' : X \to Y$ is a morphism of class $C^r$ mapping $x$ onto $y$, we have $\varphi_* = \varphi'_*$ if and only if $j_x^r(\varphi) = j_x^r(\varphi')$, cf. 12.1.

If $\varphi$ is an immersion (resp. a submersion) at $x$, $\varphi_*$ is injective (resp. surjective); the converse is true if $\dim_y Y < +\infty$.

If $\varphi' : Y \to Z$ is a morphism of $C^r$ manifolds, and if $t \in T_x^{(r)}(X)$, we have $(\varphi' \circ \varphi)_*(t) = \varphi'_*(\varphi_*(t))$.

Let $F$ be a separated polynormed space, and let $f$ be a function of class $C^r$ with values in $F$, defined in a neighbourhood of $y$. We then have

13.2.4. Suppose that $X$ is an open submanifold of a Banach space $E$. Denote by $\varphi_x$ the mapping $y \mapsto y - x$ of $X$ into $E$; the chart $c = (X, \varphi_x, E)$ is centred

¹ When $K = \mathbf{R}$ or $\mathbf{C}$ and $X$ is locally compact, the preceding formula leads to identifying the point distribution $\varepsilon_x$ with the Dirac measure $\varepsilon_x$ defined in JNT, III, § 1, No. 3. More generally, every measure with finite support in $X$ is identified with a distribution with finite support in the sense of No. 13.6.

at $x$. We then identify $T_x^{(r)}(X)$ with $TS^{(r)}(E)$ by means of $\theta_c^{-1}$; in particular, we have $T_x^{(\infty)}(X) = TS(E)$.

Let $F$ be a Banach space, let $u : E \to F$ be a continuous linear mapping, and let $x \in E, y \in F$ be such that $u(x) = y$. Identify as above $T_x^{(\infty)}(E)$ with $TS(E)$ and $T_y^{(\infty)}(F)$ with $TS(F)$. The mapping

13.2.5. If $k \leq r$, denote by $T^{(k)}(X)$ (resp. $T^{(k)+}(X)$) the sum set of the $T_a^{(k)}(X)$ (resp. of the $T_a^{(k)+}(X)$) for $a \in X$. A mapping $t : X \to T^{(k)}(X)$ such that $t(a) \in T_a^{(k)}(X)$ for all $a \in X$ is called a field of point distributions of order $\leq k$.

Suppose $k$ finite and $X$ locally of finite dimension. For every $a \in X$, the bilinear form $(j, t) \mapsto \langle j, t \rangle$ (cf. 13.2.2) defines an isomorphism $i_a$ of $T_a^{(k)}(X)$ onto the dual $P_a^k(X)^*$ of $P_a^k(X)$. The $i_a$ define a bijection $i : T^{(k)}(X) \to P^k(X)^*$, where $P^k(X)^*$ denotes the dual of the vector bundle $P^k(X)$; by transport of structure by means of $i^{-1}$, one endows $T^{(k)}(X)$ with a structure of vector bundle with base $X$ and of class $C^{r-k}$. A field of point distributions of order $\leq k$ is said to be of class $C^s$, with $s \leq r - k$, if it is a section of class $C^s$ of the vector bundle $T^{(k)}(X)$.

Let $\varphi : X \to Y$ be a morphism of manifolds of class $C^r$ and suppose that $Y$ is, like $X$, locally of finite dimension. Then $\varphi_* : T^{(k)}(X) \to T^{(k)}(Y)$ is a $\varphi$-morphism of vector bundles of class $C^{r-k}$.

13.2.6. Let $k$ be an integer such that $0 \leq k \leq r$. If $U$ is an open set of a Banach space $E$ of finite dimension, the vector bundle $T^{(k)}(U)$ is identified, thanks to 13.2.4, with the trivial bundle of fibre $TS^{(k)}(E)$.

Take in particular $E = K^n$, with $n \geq 0$, and let $(e_1, \ldots, e_n)$ be the canonical basis of $E$. If $\alpha = (\alpha_1, \ldots, \alpha_n)$ is an element of $\mathbf{N}^n$, denote by $\Delta^\alpha$ the element $\gamma_{\alpha_1}(e_1) \ldots \gamma_{\alpha_n}(e_n)$ of $TS(E)$ (the product used being the symmetric product of symmetric tensors, cf. A, IV, § 5, No. 3).(1) The $\Delta^\alpha$, for $|\alpha| \leq k$, form a basis of $TS^{(k)}(E)$; if $a \in K^n$, denote by $\Delta_a^\alpha$ the corresponding elements of $T_a^{(k)}(E)$. Let $F$ be a separated polynormed space, and $f$ a function of class $C^r$, with values in $F$, defined in a neighbourhood of $a$; if $\alpha \in \mathbf{N}^n$, the element $\langle f, \Delta_a^\alpha \rangle$ coincides with the element $(\Delta^\alpha f)(a)$ defined in 2.5.3, 3.2.1 and 4.2.1.

Suppose $X$ is locally of finite dimension, and let $\xi = (\xi^1, \ldots, \xi^n)$ be a system of coordinates of $X$ in an open set $U$. If $a \in U$ and if $\alpha$ is a multi-index such that $|\alpha| \leq k$, denote by $(\Delta_\xi^\alpha)_a$ the point distribution at $a$ whose image by $\xi$ is the point distribution $\Delta_{\xi(a)}^\alpha$ on $K^n$; the fields of distributions

(1) If $m = |\alpha|$ and if $\Sigma$ is the set of mappings $\sigma$ of $\{1, \ldots, m\}$ into $\{1, \ldots, n\}$ such that $\mathrm{Card}\, \sigma^{-1}(i) = \alpha_i$, we have

13.3. Point distributions and tangent spaces

In this No., $X$ denotes a variety of class $C^r$.

13.3.1. Let $x \in X$ and let $k$ be an integer $\leq r$. Let $c=(U,\varphi,E)$ be a chart of $X$ centred at $x$; the isomorphism $\theta_c:T^S(X)(E)\to T_x(X)$ defined in 13.2.1 maps $T^S{}^{(k)}(E)$ onto $T_x^{(k)}(X)$ and $T^S{}^{(k-1)}(E)$ onto $T_x^{(k-1)}(X)$; by restriction and passing to the quotient, it induces an isomorphism

On the other hand, $c$ defines an isomorphism, already denoted by $\theta_c$, of $E$ onto the tangent space $T_x(X)$, cf. 5.5.1. Let $i_k$ denote the composite

The isomorphism $i_k$ is independent of the choice of $c$; for $k=1$, it is used to identify $T_x^{(1)}(X)=T_x^{(1)}(X)/T_x^{(0)}(X)$ with the tangent space $T_x(X)$. If $t\in T_x^{(k)}(X)$, we allow ourselves to denote by $i_k(t)$ the image by $i_k$ of the class of $t$ modulo $T_x^{(k-1)}(X)$.

Put

and

We say that $\operatorname{gr}T_x^{(r)}(X)$ is the graded associated with the increasing filtration $(T_x^{(k)}(X))_{k\leq r}$ of $T_x^{(r)}(X)$. The $i_k$ define an isomorphism of graded vector spaces

If $r=\infty$ or $\omega$, $i$ is an isomorphism of $\operatorname{gr}T_x^{(r)}(X)$ onto $T^S(T_x(X))$. When one wishes to specify $x$ (or $X$, or both), one writes $i_x$ (or $i_X$, or $i_{x,X}$) in place of $i$.

13.3.2. In addition to the above hypotheses, suppose that $X$ is locally finite-dimensional. The isomorphisms $i_k$ corresponding to the different points of $X$ define an isomorphism of vector bundles

where $TS^{(k)}(T(X))$ denotes the vector bundle of class $C^{r-1}$ obtained from $T(X)$ by the finite-dimensional vector functor $TS^{(k)}$ (cf. 7.6.5). For $k\geq 1$, $i_k$ is an isomorphism of class $C^{r-k}$; for $k=0$, it is the identity mapping of the trivial bundle $K_X$.

13.3.3. Example. — The hypotheses being as above, let $\xi=(\xi^1,\ldots,\xi^n)$ be a system of coordinates of $X$ at $x$. Let $(\partial_i)_{1\leq i\leq n}$ be the basis of $T_x(X)$ defined by $\xi$ (cf. 5.5.8) and let $(\Delta^\alpha_x)_{|\alpha|\leq k}$ be that of $T_x^{(k)}(X)$ defined in 13.2.6. We have:

[^1]

13.3.4. Suppose that K is of characteristic zero or that X is locally of finite dimension. Let k be an integer such that $0 \leq k \leq r$ and let $x \in X$; let F be a Banach space. By 12.6.8, we have an exact sequence

(i)

(ii)

(i)

(ii)

If $a \in A$ and $a' \in A'$, the element $\langle a, a' \rangle$ of F is defined by 13.1.2; if $b \in B$ and $b' \in B'$, (resp. if $c \in C$ and $c' \in C'$), the element $\langle b, b' \rangle$ (resp. $\langle c, c' \rangle$) of F is defined by 13.2.2; we then have:

13.3.5. Let Y be a variety of class $C^r$, let $\varphi : X \to Y$ be a morphism of class $C^r$ and let $x \in X,\ y \in Y$ be such that $y = \varphi(x)$. The mapping $\varphi_* : T_x^{(r)}(X) \to T_y^{(r)}(Y)$ defined in 13.2.3 is compatible with the filtrations of these spaces; it defines by passing to the associated graded spaces a linear mapping

is commutative.

13.3.6. Suppose that K is of characteristic zero. Using the isomorphisms $\varphi_M : S^k(M) \to TS^k(M)$ defined in A, IV, § 5, No. 8, we can replace, in all that precedes, the $TS^k(T_x(X))$ by the k-th symmetric powers $S^k(T_x(X))$ of the tangent space $T_x(X)$.

13.4. Tensor product of point distributions

13.4.1. Let $X_1$ and $X_2$ be two manifolds of class $C^r$, let $x_1 \in X_1,\ x_2 \in X_2$ and $t_1 \in T_{x_1}^{(k_1)}(X_1),\ t_2 \in T_{x_2}^{(k_2)}(X_2)$ with $k_1 + k_2 \leq r$. Put $X = X_1 \times X_2,\ x = (x_1, x_2)$ and k = k_1 + k_2. For i = 1, 2, let c_i = (U_i, \varphi_i, E_i) be a chart of X_i centred at x_i and denote by $\tilde{t}_i$ the element of TS(E_i) such that $\theta_c(\tilde{t}_i) = t_i$, cf. 13.2.1. Let $\sigma$ be the canonical isomorphism of TS(E_1) $\otimes$ TS(E_2) onto TS(E_1 $\times$ E_2), cf. A, IV, § 5, no. 5. The element $\sigma(\tilde{t}_1 \otimes \tilde{t}_2)$ is the symmetric product of $\tilde{t}_1$ and $\tilde{t}_2$ (identified with elements of TS(E_1 $\times$ E_2) by means of the canonical injections TS(E_i) $\to$ TS(E_1 $\times$ E_2)); it belongs to TS^{(k)}(E_1 $\times$ E_2). Put $c = c_1 \times c_2$; this is a chart of X centred at x. The image by $\theta_c$ of $\sigma(\tilde{t}_1 \otimes \tilde{t}_2)$ is an element of T_x^{(k)}(X), which does not depend on the choice of the charts $c_i$. It is called the direct product, or the symmetric tensor product (or simply the tensor product) of $t_1$ and $t_2$ and is denoted by $t_1 \times t_2$ or $t_1 \otimes t_2$.

We have $\varepsilon_{x_1} \otimes \varepsilon_{x_2} = \varepsilon_x$. The constant term of $t_1 \otimes t_2$ is the product of the constant terms of $t_1$ and $t_2$.

The isomorphism $(y_1, y_2) \mapsto (y_2, y_1)$ of $X_1 \times X_2$ onto $X_2 \times X_1$ transforms $t_1 \otimes t_2$ into $t_2 \otimes t_1$.

13.4.2 (« Associativity and functoriality »). Let $X_i$ ($i = 1, 2, 3$) be manifolds of class $C^r$, and let $x_i \in X_i$, $t_i \in T^{(k_i)}_{x_i}(X_i)$ with $k_1 + k_2 + k_3 \leq r$. We have

Let $\varphi_1 : X_1 \to Y_1$ and $\varphi_2 : X_2 \to Y_2$ be morphisms of manifolds of class $C^r$, and let $t_1 \in T^{(k_1)}(X_1)$, $t_2 \in T^{(k_2)}(X_2)$ with $k_1 + k_2 \leq r$. We have

13.4.3. The notation being that of 13.4.1, let $F_1, F_2$ and F be separated polynormed spaces, and let $(u_1, u_2) \mapsto u_1 . u_2$ be a continuous bilinear mapping of $F_1 \times F_2$ into F. Let

13.4.4. The notation being that of 13.4.1, let $f$ be a mapping of class $C^r$ of X into a separated polynormed space F. If $y_1 \in X_1$, denote by $f_{y_1}$ the mapping $y_2 \mapsto f(y_1, y_2)$ of $X_2$ into F and put $g(y_1) = \langle f_{y_1}, t_2 \rangle$. The function $g : X_1 \to F$ thus defined is of class $C^{r - k_1}$, and we have