# Jet Isomorphism

# Jet Isomorphism

# Abstract and Keywords

A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors. This chapter proves an analogous jet isomorphism theorem for conformal geometry. By making conformal changes, the Taylor expansion of a metric in geodesic normal coordinates can be further simplified, resulting in a “conformal normal form” for metrics about a point.

*Keywords:*
jet isomorphism theorem, conformal geometry, ambient metric, Riemannian geometry, geodesic normal coordinates, Taylor expansion

A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors.

Our goal in this chapter is to prove an analogous jet isomorphism theorem for conformal geometry. By making conformal changes, the Taylor expansion of a metric in geodesic normal coordinates can be further simplified, resulting in a "conformal normal form" for metrics about a point. The jet isomorphism theorem states that the map from the Taylor coefficients of metrics in conformal normal form to the space of all conformal curvature tensors, realized in terms of covariant derivatives of ambient curvature, is an isomorphism. If *n* is even, the theorem holds only up to a finite order. In the conformal case, the role of the orthogonal group in Riemannian geometry is played by a parabolic subgroup of the conformal group. We assume throughout this chapter and the next that *n* ≥ 3.

We begin by reviewing the Riemannian theorem in the form we will use it.
Fix a reference quadratic form *h _{ij}* of signature (

*p, q*) on ${\mathbb{R}}^{n}.$ . In the positive definite case one typically chooses

*h*=

_{ij}*δ*

_{ij}.The background coordinates are geodesic normal coordinates for a metric

*g*defined near the origin in ${\mathbb{R}}^{n}$ if and only if

_{ij}*g*(0)=

_{ij}*h*and the radial vector field

_{ij}*∂*satisfies ${\nabla}_{{\partial}_{r}}{\partial}_{r}=0,$ which is equivalent to ${\Gamma}_{jk}^{i}{x}^{j}{x}^{k}=0$ or

_{r}It is easily seen that the coordinates are normal if and only if

In fact, set ${F}_{i}={g}_{ij}{x}^{j}-{h}_{ij}{x}^{j}$ and observe that

(p.83)
Clearly *F _{i}* = 0 implies that (8.1) holds so that the coordinates are normal. Conversely, if the coordinates are normal, then the fact that

*g*and

*∂*

*are parallel along radial lines implies ${x}^{j}{F}_{j}=0.$ Substituting this and (8.3) into (8.1) gives ${x}^{j}{\partial}_{j}{F}_{j}=0,$ , which together with the fact that*

_{r}*F*is smooth and vanishes at the origin implies

_{i}*F*= 0.

_{i}We are interested in the space of infinite order jets of metrics at the origin. Taylor expanding shows that (8.2) holds to infinite order if and only if *g _{ij}* (0)=

*h*and the derivatives of

_{ij}*g*satisfy ${\partial}_{({k}_{1}}\mathrm{...}{\partial}_{{k}_{r}gi)j}(0)=0.$ Since a 3-tensor symmetric in 2 indices and skew in 2 indices must vanish, this implies in particular that all first derivatives of

_{ij}*g*vanish at the origin. So we define the space $\mathcal{N}$ of jets of metrics in geodesic normal coordinates as follows.

Definition 8.1.

The space $\mathcal{N}$ is the set of lists (

g^{(2)},g^{(3)}, …), where for each $r,{g}^{(r)}\in {\odot}^{2}{\mathbb{R}}^{n\ast}\otimes {\odot}^{r}{\mathbb{R}}^{n\ast}$ satisfies ${g}_{i(j,{k}_{1}\cdots {k}_{r})}^{(r)}=0$ Here the comma just serves to separate the first two indices. For $N\ge 2,{\mathcal{N}}^{N}$ will denote the set of truncated lists (g^{(2)},g^{(3)},…,g^{(N)}) with the same conditions on theg^{(r)}.To a metric in geodesic normal coordinates near the origin, we associate the element of $\mathcal{N}$ given by ${g}_{ij,{k}_{1}\mathrm{...}{k}_{r}}^{(r)}={\partial}_{{k}_{1}}\mathrm{...}{\partial}_{{k}_{r}{g}_{ij}}(0)$ for

r≥ 2. Conversely, to an element of $\mathcal{N}$ we associate the metric determined to infinite order by these relations together withg(0)=_{ij}hand_{ij}∂_{k}g(0)= 0. In the following, we typically identify the element of $\mathcal{N}$ and the jet of the metric_{ij}g.

Definition 8.2.

The space $\mathcal{R}$ is the set of lists (

R^{(0)},R^{(1)},…), where for each $r,{R}^{(r)}\in \text{}\text{}{\wedge}^{2}{R}^{n\ast}\otimes \text{}{\wedge}^{2}{R}^{n\ast}\otimes {\otimes}^{r}{R}^{n\ast},$ and the usual identities satisfied by covariant derivatives of curvature hold:

(1) ${R}_{i[jkl],{m}_{1}\cdots {m}_{r}}=0$;

(2) ${R}_{ij[kl,{m}_{1}]{m}_{2}\cdots {m}_{r}}=0$;

(3) ${R}_{ijkl,{m}_{1}\cdots [{m}_{s-1}{m}_{s}]\cdots {m}_{r}}={Q}_{ijkl{m}_{1}\cdots {m}_{r}}^{(s)}$.

Here ${Q}_{ijkl{m}_{1}\cdots {m}_{r}}^{(s)}$ denotes the quadratic polynomial in the ${R}^{({r}^{\prime})}$ with

r'≤r– 2 which one obtains by covariantly differentiating the usual Ricci identity for commuting covariant derivatives, expanding the differentiations using the Leibnitz rule, and then setting equal tohthe metric which contracts the two factors in each term.

For $N\ge 0,{\mathcal{R}}^{N}$ will denote the set of truncated lists (*R*^{(0)}, *R*^{(1)},…,*R*^{(N)}) with the same conditions on the *R*^{(r)}.

There is a natural map $\mathcal{N}\to \mathcal{R}$ induced by evaluation of the covariant derivatives of curvature of a metric, which is polynomial in the sense that (p.84) the corresponding truncated maps ${\mathcal{N}}^{N+2}\to {\mathcal{R}}^{N}$ are polynomial. We will say that a map on a subset of a finite-dimensional vector space is polynomial if it is the restriction of a polynomial map defined on the whole space, and a map between subsets of finite-dimensional vector spaces is a polynomial equivalence if it is a bijective polynomial map whose inverse is also polynomial. The jet isomorphism theorem for pseudo-Riemannian geometry is the following.

Theorem 8.3.

The map $\mathcal{N}\to \mathcal{R}$ is bijective and for each Ν≥0the truncated map${\mathcal{N}}^{N+2}\to {\mathcal{R}}^{N}$is a polynomial equivalence.

There are two parts to the proof. One is a linearization argument showing that it suffices to show that the linearized map is an isomorphism. The other is the observation that the linearized map is the direct sum over *r* of isomorphisms between two realizations according to different Young projectors of specific irreducible representations of $GL(n,\mathbb{R}).$ The reduction to the linearization can be carried out in different ways; one is to argue as we do below in the conformai case. The analysis of the linearized map is contained in [E2]. See also [ABP] for a different construction of a left inverse of the map
$\mathcal{N}\to \mathcal{R}.$ .

We now consider the further freedom allowed by the possibility of making conformal changes to *g.*

Proposition 8.4.

Let g be a metric on a manifold M and let p ∈ M. Given${\Omega}_{0}\in {C}^{\infty}(M)$with${\Omega}_{0}(p)>0,$there is$\Omega \in {C}^{\infty}(M),$uniquely determined to infinite order at p, such thatΩ − Ω_{0}vanishes to second order at p and$\text{Sym(}{\nabla}^{r}\text{Ric)(}{\Omega}^{2}g)(p)=0$for allr≥ 0.Here$\text{Sym(}{\nabla}^{r}\text{Ric)(}{\Omega}^{2}g)$denotes the full symmetrization of the rank r+ 2tensor$\text{(}{\nabla}^{r}\text{Ric)(}{\Omega}^{2}g).$

Proof.

Write $\Omega ={e}^{\Upsilon}$ and set $\widehat{g}={e}^{2\Upsilon}g.$ We are given $\Upsilon (p)$ and $\text{}d\Upsilon (p).$ . Recall the transformation law for conformal change of Ricci curvature:

$${\widehat{R}}_{ij}={R}_{ij}-(n-2){\Upsilon}_{ij}-{\Upsilon}_{k}{}^{k}{g}_{ij}+(n-2)({\Upsilon}_{i}{\Upsilon}_{j}-{\Upsilon}_{k}{\Upsilon}^{k}{g}_{ij}).$$Differentiating and conformally transforming the covariant derivative results in

$$\begin{array}{l}{\left(\widehat{{\nabla}^{r}\text{Ric}}\right)}_{ij,{m}_{1}\mathrm{...}{m}_{r}}={(}_{{\nabla}^{r}}-(n-2){\Upsilon}_{ij{m}_{1}\mathrm{...}{m}_{r}}\\ \text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}-{\Upsilon}_{k}{{}^{k}}_{{m}_{1}\mathrm{...}{m}_{r}}{g}_{ij}+\text{lots,}\end{array}$$where lots denotes terms involving at most

r+ 1 derivatives of $\Upsilon .$ . We may replace covariant derivatives of $\Upsilon $ by coordinate derivatives on the right-hand side of this formula. Then symmetrizing gives$$\text{Sym}\text{}\text{(}\widehat{{\nabla}^{r}\text{Ric}}\text{)=Sym}\text{}\text{(}{\nabla}^{r}\text{Ric)}\text{}-\text{(}n-\text{2)}{\partial}^{r+2}\Upsilon -\text{Sym}\text{}(\text{tr}({\partial}^{r+2}\Upsilon )g)+\text{lots}\text{.}$$(p.85) If

c> 0, the map $s\to s+c\text{}\text{Sym(tr(}s\text{)}g)$ is invertible on symmetric (r+ 2)- tensors. Therefore, for anyr≥ 0, one can uniquely determine ${\partial}^{r+2}\Upsilon (p)$ to make $\text{Sym}\text{}\text{(}\widehat{{\nabla}^{r}\text{Ric}}\text{)(}p\text{)=0}\text{.}$ The result follows upon iterating and applying Borel's Lemma.

We remark that there are other natural choices for normalizations of the conformal factor. For example, one such is that the symmetrized covariant derivatives of the tensor *P _{ij}* vanish at

*p*, where

*P*is given by (3.7). Another is that in normal coordinates, det

_{ij}*g*− 1 vanishes to infinite order at

_{ij}*p*(see [LP]).

Definition 8.5.

The space ${\mathcal{N}}_{c}\subset \mathcal{N}$ of jets of metrics in conformal normal form is the subset of $\mathcal{N}$ consisting of jets of metrics in geodesic normal coordinates for which $\text{Sym}\text{}\text{(}{\nabla}^{r}\text{Ric})\text{(}g\text{)(0)=0}$ for all

r≥ 0. ForΝ≥ 2, ${\mathcal{N}}_{c}^{N}$ will denote the subset of ${\mathcal{N}}^{N}$ obtained by requiring that this relation hold for 0 ≤r≤N− 2.Later we will need the following consequence of the proof of Proposition 8.4.

Lemma 8.6.

For each Ν≥ 3,there is a polynomial map${\eta}^{N}:{\mathcal{N}}_{c}^{N-1}\to {\odot}^{2}{\mathbb{R}}^{n\ast}\text{}\otimes {\odot}^{N}{\mathbb{R}}^{n\ast}\text{}$with zero constant and linear terms, such that the map$g\to \text{}\text{}(g,{\eta}^{N}(g))$maps${\mathcal{N}}_{c}^{N-1}\to {\mathcal{N}}_{c}^{N}.$Here$g=({g}^{(2)},{g}^{(3)},\mathrm{...},{g}^{(N-1)})\in {\mathcal{N}}_{c}^{N-1}.$

Proof.

View

gas the metric on a neighborhood of $0\in {\mathbb{R}}^{n}$ given by the prescribed finite Taylor expansion of orderΝ– 1. Then the components of $\text{Sym(}{\nabla}^{N-2}\text{Ric)(}g\text{)(0)}$ are polynomials in theg^{(r)}with no constant or linear terms. According to the proof of Proposition 8.4, there is a function Ω of the form Ω = 1 +pwith_{Ν}pa homogeneous polynomial of degree_{Ν}Νwhose coefficients are polynomials in theg^{(r)}with no constant or linear terms, such that $\text{Sym(}{\nabla}^{N-2}\text{Ric)}({\Omega}^{2}g)(0)=0.$ Now Ω^{2}gis in geodesic normal coordinates to orderΝ– 1 but not necessarilyN.However by the construction of geodesic normal coordinates, there is a diffeomorphism ψ = I +q_{N+1}, whereq_{N+1}is a vector-valued homogeneous polynomial of orderN+l whose coefficients are linear in the orderΝTaylor coefficient of Ω^{2}g, such that ${\psi}^{\ast}({\Omega}^{2}g)$ is in geodesic normal coordinates to orderN.Since the condition $\text{Sym(}{\nabla}^{r}\text{Ric)}({\Omega}^{2}g)\text{(0)=0}$ is invariant under diffeomorphisms,ηdefined to be the order^{Ν}ΝTaylor coefficient of ${\psi}^{\ast}({\Omega}^{2}g)$ has the required properties.

If *g _{1}* and

*g*are jets of metrics of signature (

_{2}*p, q*) at the origin in ${\mathbb{R}}^{n},$, we say that

*g*and

_{1}*g*are equivalent if there is a local diffeomorphism

_{2}*ψ*defined near 0 satisfying

*ψ*(0) = 0,and a positive smooth function Ω defined near 0, so that ${g}_{2}={\psi}^{\ast}({\Omega}^{2}{g}_{1})$ to infinite order. It is clear from Proposition 8.4 (p.86) and the existence of geodesic normal coordinates that any jet of a metric is equivalent to one in

*N*In choosing Ω we have the freedom of ${\mathbb{R}}_{+}\times {\mathbb{R}}^{n},$ and in choosing

_{c}.*ψ*a freedom of

*O*(

*p,q*). We next describe how these freedoms can be realized as an action on

*N*of a subgroup of the conformai group.

_{c}Recall that *h _{ij}* is our fixed background quadratic form of signature (

*p,q*) on ${\mathbb{R}}^{n}.$ . Define a quadratic form ${\tilde{h}}_{IJ}$ on ℝ

^{n}+

^{2}by

and the quadric $\mathcal{Q}=\{[{x}^{I}]:{\tilde{h}}_{IJ}{x}^{I}{x}^{J}=0\}\subset {\mathbb{P}}^{n+1}.$ The metric ${\tilde{h}}_{IJ}d{x}^{I}d{x}^{J}$ on ℝ^{n+2} induces a conformal structure of signature (*p,q*) on *Q.* The standard action of the orthogonal group $O(\tilde{h})$ on ℝ^{n+2} induces an action on *Q* by conformal transformations and the adjoint group $O(\tilde{h})/\{\pm I\}$ can be identified with the conformal group of all conformal transformations of *Q.*

Let ${e}_{0}=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)\in {\mathbb{R}}^{n+2}$ and let *Ρ* be the image in $O(\tilde{h})/\{\pm I\}$ of the isotropy group $\{A\in O(\tilde{h}):A{e}_{0}=a{e}_{0},a\in \mathbb{R}\}\text{}\text{}\text{}\text{of}\text{}\text{}\text{[}{e}_{0}].$ It is clear that each element of *Ρ* is represented by exactly one *A* for which *a* > 0, so we make the identification $P=\{p\in O(\tilde{h}):p{e}_{0}=a{e}_{0},\text{}a>0\}.$ The first column of $p\in P$ is $\left(\begin{array}{c}a\\ 0\\ 0\end{array}\right);$ combining this with the fact that $p\in O(\tilde{h}),$ one finds that

where *h _{ij}* is used to raise and lower lowercase indices. It is evident that

*Ρ*= ℝ

_{+}· ℝ

^{n}·

*Ο*(

*h*), where the subgroups ℝ

_{+}, ℝ

^{n},

*Ο*(

*h*) arise by varying

*a*,

*b*, resp.

_{j}, m^{i}_{j}The intersection of $\mathcal{Q}$ with the cell $\{[{x}^{I}]:{x}^{0}\ne 0\}$ can be identified with ℝ^{n} via ${\mathbb{R}}^{n}\ni {x}^{i}\to \left[\begin{array}{c}1\\ {x}^{i}\\ -\frac{1}{2}|x{|}^{2}\end{array}\right]\in \mathcal{Q},$ where $|x{|}^{2}={h}_{ij}{x}^{i}{x}^{j}.$ In this identification, the conformal structure is represented by the metric ${h}_{ij}d{x}^{i}d{x}^{j}$ on ℝ^{n}. If *p* ∈ *Ρ* is as above, the conformal transformation determined by *p* is realized as

and one has ${\phi}_{p}^{\ast}h={\mathrm{\Omega}}_{p}^{2}h$ for

This motivates the following definition of an action of Ρ on ${\mathcal{N}}_{c}$. Given
(p.87)
$p\in P$ as above and $g\in {\mathcal{N}}_{c},$ by Proposition 8.4 there is a positive smooth function Ω uniquely determined to infinite order at 0 so that Ω agrees with Ω_{p} to second order and such that $\text{Sym}({\nabla}^{r}\text{Ric})({\text{\Omega}}^{2}g)(0)=0$ for all r > 0. Now $({\text{\Omega}}^{2}g)(0)={a}^{-2}h,$ so by the construction of geodesic normal coordinates, there is a diffeomorphism *φ*, uniquely determined to infinite order at 0, so that $\phi (0)=0,\text{}\text{}\phi \text{'}(0)={a}^{-1}m,$ and such that ${({\phi}^{-1})}^{\ast}({\text{\Omega}}^{2}g)$ is in geodesic normal coordinates to infinite order. We define $p.g={({\phi}^{-1})}^{\ast}({\text{\Omega}}^{2}g).$ . It is clear by construction of *φ* that $p.g\in \mathcal{N},$ and since the condition of vanishing of the symmetrized covariant derivatives of Ricci curvature is diffeomorphism-invariant, it follows by construction of Ω that $p.g\in {\mathcal{N}}_{c}.$ It is straightforward to check that this defines a left action of Ρ on ${\mathcal{N}}_{c}$. Note that if *g = h*, then $\text{\Omega}={\text{\Omega}}_{p}$ and $\phi ={\phi}_{p}$ so that ft is a fixed point of the action. A moment's thought shows that *(p.g) ^{(r)}* depends only on

*g*for

^{(s)}*s < r.*Therefore for each $N\ge 2,$ there is an induced action on ${\mathcal{N}}_{c}^{N}.$

It is clear from the construction of the action that *p.g* is equivalent to *g* for all $g\in {\mathcal{N}}_{c}$ and $p\in P$ In fact, the P-orbits are exactly the equivalence classes.

Proposition 8.7.

The orbits of the Ρ-action on ${\mathcal{N}}_{c}$ are precisely the equivalence classes of jets of metrics in ${\mathcal{N}}_{c}$ under diffeomorphism and conformai change.

Proof. It remains to show that equivalent jets of metrics in ${\mathcal{N}}_{c}$ in the same P-orbit. Suppose that ${g}_{1},{g}_{2}\in {\mathcal{N}}_{c}$ are equivalent. Then we can write ${g}_{2}={(}^{{\phi}^{-1}}({\text{\Omega}}^{2}{g}_{1})$ to infinite order for a diffeomorphism

φ with $\phi (0)=0$ and a positive smooth Ω. We can uniquely choose the parametersaand b of $p\in P$ so that $\mathrm{\Omega}-{\mathrm{\Omega}}_{p}$ vanishes to second order. Sinceg1andg2both equalhat 0, it follows that $a{\phi}^{\prime}(0)\in O(h),$ so that we can write ${\phi}^{\prime}(0)={a}^{-1}m$ with $m\in O(h).$ Together with the already determined parametersaandb, this choice of m uniquely determines a $p\in P$ Since ${g}_{2}\in {\mathcal{N}}_{c},$ all symmetrized covariant derivatives of Ricci curvature of ${\Omega}^{2}{g}_{1}$ vanish at the origin, so Ω must be the conformai factor determined when constructing the action ofρong_{1}. And sinceg_{1}is in geodesic normal coordinates, φ must be the correct diffeomorphism, so thatg_{2}=p.g_{1}.It is straightforward to calculate from the definition the action on ${\mathcal{N}}_{c}$ of the ℝ

_{+}andO(h)subgroups ofP. If we denote bypthe element of_{a}Pobtained by taking b = 0 andm = Iand bypthe element given by a =1 and b = 0, then one finds that_{m}pacts by multiplying_{a}g_{(r)}byaand_{r}pacts by transforming each_{m}g_{(r)}as an element of ${\otimes}^{r+2}{\mathbb{R}}^{n\ast},$ where ℝ_{n}denotes the standard defining representation ofΟ(h).(p.88) The problem of understanding local invariants of metrics under diffeomorphism and conformal change reduces to understanding this action of

Ρon N^{C}. However, it is very difficult to analyze or even concretely exhibit the action of the ℝ^{n}part ofΡdirectly from the definition. The ambient curvature tensors enable the reformulation of the action in terms of standard tensor representations ofP.Propositions 6.1 and 6.4 show that the covariant derivatives of curvature of an ambient metric $\tilde{g}$ satisfy relations arising from the homogeneity and Ricci-flatness of the metric. These conditions suggest the following definition.

Definition 8.8.The space $\tilde{\mathcal{R}}$ is the set of lists $({\tilde{R}}^{(0)},{\tilde{R}}^{(1)},\mathrm{...}),$ such that ${\tilde{R}}^{(r)}\in {\wedge}^{2}{\mathbb{R}}^{n+2}{}^{\ast}\otimes {\wedge}^{2}{\mathbb{R}}^{n+2}{}^{\ast}\otimes {\otimes}^{r}{\mathbb{R}}^{n+2}{}^{\ast},$ and such that the following relations hold:

(1) ${\tilde{R}}_{I[JKL],{M}_{1}\cdots {M}_{r}}=0;$

(2) ${\tilde{R}}_{IJ[KL,{M}_{1}]{M}_{2}\cdots {M}_{r}}=0;$

(3) ${\tilde{h}}^{IK}{\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}=0;$

(4) ${\tilde{R}}_{IJKL,{M}_{1}\cdots [{M}_{s-1}{M}_{s}]\cdots {M}_{r}}={\tilde{Q}}_{IJKL{M}_{1}\cdots {M}_{r}}^{(s)};$

(5) ${\tilde{R}}_{IJK0,{M}_{1}\cdots {M}_{r}}=-{\displaystyle {\sum}_{s=1}^{r}{\tilde{R}}_{IJK{M}_{s},{M}_{1}\cdots \widehat{{M}_{s}}\mathrm{...}{M}_{r};}}$

(6) $\begin{array}{l}{\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{s}0{M}_{s+1}\cdots {M}_{r}}=\\ \text{}\text{}-(s+2){\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}-{{\displaystyle {\sum}_{t=s+1}^{r}\tilde{R}}}_{IJKL,{M}_{1}\cdots {M}_{s}{M}_{t}{M}_{s+1}\cdots \widehat{{M}_{t}}\cdots {M}_{r}}.\end{array}$

Here, as in Definition 8.2, ${\tilde{Q}}_{IJKL{M}_{1}\cdots {M}_{r}}^{(s)}$ denotes the quadratic polynomial in the components of ${\tilde{R}}^{({r}^{\prime})}\text{}\text{}\text{for}\text{}\text{}{r}^{\prime}\le r-2$ which one obtains by differentiating the Ricci identity for commuting covariant derivatives, expanding the differentiations using the Leibnitz rule, and then setting equal to $\tilde{h}$ the metric which contracts the two factors in each term. Condition (5) in case r = 0 is interpreted as ${\tilde{R}}_{IJK0}=0.$

We remark, as we did in the proof of Proposition 6.1, that condition (6) is superfluous: it is a consequence of (2), (4), (5). But we

will not use this fact.

It will be convenient also to introduce the vector space $\tilde{\tau}$ consisting of the set of lists $({\tilde{R}}^{(0)},{\tilde{R}}^{(1)},\dots )$ with ${\tilde{R}}^{(r)}\in {\wedge}^{2}{\mathbb{R}}^{n+2\ast}\otimes {\wedge}^{2}{\mathbb{R}}^{n+2\ast}\otimes {\otimes}^{r}{\mathbb{R}}^{n+2\ast}$ such that (l)–(3), (5), (6) hold.

We prepare to define truncated spaces ${\tilde{\mathcal{R}}}^{N}$ for $\tilde{\mathcal{R}}.$ Recall the notion of

strength from Définition 6.3. Note that it is clear that for each of the relations (l)–(6) except (4) in Definition 8.8, all components ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ which occur in the relation have the same value for the strength of the index list
(p.89)
of the component. For *Ν >* 0, define the following vector spaces of lists of components of tensors. We denote by a list of indices of length $\left|\mathcal{M}\right|=r,$ and the conditions (l)–(6) refer to Definition 8.8.

If is a component appearing in an element of ${\tilde{\tau}}^{N}$ and *r > N*, then at least one of the indices *IJKLM _{1} ... M_{r}* must be 0. Therefore by (5) and (6), ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ can be written as a linear combination of

components with *r* replaced by *r* – 1. It follows that
${\tilde{\tau}}^{N}\text{}\text{}\text{and}\text{}\text{}{\tilde{\tau}}^{N}$

are finite-dimensional. Since (l)–(3), (5) and (6) imply that ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}=0$ if $\parallel IJKL{M}_{1}\cdots {M}_{r}\parallel \le 3,$ we have ${\tilde{T}}^{N}={\oplus}_{M=0}^{N}{\tilde{T}}^{M}$and $\tilde{T}={\displaystyle \prod {}_{M=0}^{\infty}}{\tilde{T}}^{M}.$ As for (4), a typical term in ${\tilde{Q}}_{IJKL{M}_{1}\cdots {M}_{r}}^{(s)}$ is

where *M4'* and *M4"* are lists of indices such that ${\mathcal{M}}^{\prime}{\mathcal{M}}^{\u2033}$ is a rearrangement of ${M}_{s+1}\cdots {M}_{r}.$ In order that ${\tilde{h}}^{AB}\ne 0,$ it must be that $\parallel AB\parallel =2.$ There-fore $\parallel AI{M}_{s-1}{M}_{s}{\mathcal{M}}^{\prime}\parallel +\parallel BJKL{M}_{1}\cdots {M}_{s-2}{\mathcal{M}}^{\u2033}\parallel =\parallel IJKL{M}_{1}\cdots {M}_{r}\parallel +2.$

This implies that if either $\parallel AI{M}_{s-1}{M}_{s}{\mathcal{M}}^{\prime}\parallel \text{}\text{}\text{or}\text{}\text{}\parallel BJKL{M}_{1}\cdots {M}_{s-2}{\mathcal{M}}^{\u2033}\parallel $ is greater than or equal to $\parallel IJKL{M}_{1}\cdots {M}_{r}\parallel ,$ then the other is less than

or equal to 2. The same reasoning applies to all terms in ${\tilde{Q}}_{IJKL{M}_{1}\cdots {M}_{r}}^{(s)}.$ Since (l)–(3), (5) and (6) imply that a component of an ${\tilde{R}}^{(t)}$ vanishes if its

strength is at most 3, it follows that any component of an ${\tilde{R}}^{(t)}$ which occurs in ${\tilde{Q}}_{IJKL{M}_{1}\cdots {M}_{r}}^{(s)}$ with a nonzero coefficient must have strength strictly less$\Vert IJKL{M}_{1}\cdots {M}_{r}\Vert .$Hence we will regard (4) as a relation involving components with indices of strength at most $\parallel IJKL{M}_{1}\cdots {M}_{r}\parallel ,$ and the quadratic terms in (4) only involve components of strength less than that of the linear terms. With this understanding, we can now define

${\tilde{\mathcal{R}}}^{N}=\{({\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}})\in {\tilde{\tau}}^{N}$ : (4) of Definition 8.8 holds}.

We will also need the corresponding linearized spaces, in which the term ${\tilde{Q}}_{IJKL{M}_{1}\cdots {M}_{r}}^{(s)}$ in (4) is replaced by 0. Define vector spaces

Then $T{\tilde{\mathcal{R}}}^{N}={\oplus}_{M=0}^{N}\text{}{\tilde{\sigma}}^{M}\text{}\text{}\text{and}\text{}\text{}T\tilde{\mathcal{R}}={\displaystyle {\prod}_{M=0}^{\infty}{\tilde{\sigma}}^{M}}.$ Note that in the presence of the condition ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}={\tilde{R}}_{IJKL,({M}_{1}\cdots {M}_{r})},$ condition (6) of Definition 8.8 becomes

and is a consequence of (2) and (5). Note also that if $({\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}})\in {\tilde{\sigma}}^{N},$

then ${\tilde{h}}^{L{M}_{s}}{\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}=0$ and ${\tilde{h}}^{{M}_{s}{M}_{t}}{\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}=0.$

For $w\in \u2102,$ Let ${\sigma}_{w}:P\to \u2102$ denote the character Since Ρ C *Ο (h)* and Ρ preserves e_{0} up to scale, it is easily seen that and $\tilde{T},\tilde{R}\text{}and\text{}T\tilde{R}$are invariant subsets of the P-space ${\prod}_{r=0}^{\infty}\left({\otimes}^{4+r}{\mathbb{R}}^{n+2\ast}\otimes {\sigma}_{-2-r}\right)},$ where

R_{n}+_{2}2 denotes the standard representation of $P\subset GL(n+2,\mathbb{R}).$ These inclusions therefore define actions of *Ρ* on these spaces. These actions of *Ρ* do not preserve strength, but because *Ρ* consists of block upper-triangular ma-trices,a component of p.(eR) depends only on components of R of greater trices, a component of *p.* depends only on strength. So for there are also actions of *Ρ* on ${\tilde{\mathcal{T}}}^{N},{\tilde{\mathcal{R}}}^{N}$ and *'*

An easy computation shows that the element acts by multiplying a

component of strength *S* by ${a}^{S-2}.$

We next define our main object of interest. If *g* is a metric defined in a neighborhood of $0\in {\mathbb{R}}^{n},$ we construct a straight ambient metric in normal form for *g* as in Chapter 3. We then evaluate the covariant derivatives of curvature of the ambient metric at $\rho =0,$

$t=1$ as described in Chapter 6. If *n* is odd, the values of all components of these covariant derivatives at the origin

depend only on the derivatives of *g* at the origin, while if *η* is even, this is true for components of strength at most n+ 1 by Proposition 6.2. If g_{ij}(0)=h_{ij},then ${\tilde{g}}_{IJ}={\tilde{h}}_{IJ}\text{}\text{}\text{at}\text{}\text{}\rho =0,\text{}\text{}t=1,\text{}\text{}x=0.$ In this case, Propositions 6.1 and 6.4 show that the resulting lists of components satisfy the relations of Definition 8.8. This procedure therefore defines a map c : for *n* odd, and $c:{\mathcal{N}}_{c}\to {\tilde{\mathcal{R}}}^{n-3}$

for *n* even. Since the conformai curvature tensors are natural polynomial invariants of the metric *g, c* is a polynomial map.

Proposition 8.9.

The map$c:{\mathcal{N}}_{c}\to \tilde{\mathcal{R}}$(or${\tilde{R}}^{n-3}$ ifnis even)is equivariant with respect to the P-actions.

Proof.

Recall that the action of

Ρ onNis given by $p.g={({\phi}^{-1})}^{\ast}({\text{\Omega}}^{2}g),$ where_{c}φand Ω are determined to mapgback to conformai normal form given the initial normalizations defined byp; see the discussion preceding Proposition 8.7 above. By naturality of the conformai curvature tensors,$c({({\phi}^{-1})}^{\ast}({\text{\Omega}}^{2}g))={({\phi}^{\prime}{(0)}^{-1})}^{\ast}(c({\text{\Omega}}^{2}g)),$ where ${({\phi}^{\prime}{(0)}^{-1})}^{\ast}$ on the right-hand side is interpreted as the pullback in the indices between 1 and

nof each of the tensors in the list, leaving the 0 and ∞ indices alone. And $c({\text{\Omega}}^{2}g)$ is given by Proposition 6.5. We use these observations to check for each of the generating subgroupsO(h), ℝ_{+}+ and ℝ_{n}ofΡthat where

the P-action on the right-hand side is that on ${\prod}_{r=0}^{\infty}\left({\otimes}^{4+r}{\mathbb{R}}^{n+2}{}^{\ast}\otimes {\sigma}_{-2-r}\right).$

For *p = p _{m}*, we have $\mathrm{\Omega}=1$ and $\phi (x)=mx,\text{}\text{}\text{so}\text{}\text{}c({p}_{m}.g)={({m}^{-1})}^{\ast}c(g)$ is obtained from

*c(g)*by transforming covariantly

under*Ο(h)* the indices

between 1 and *n*. But this is precisely how *p _{m}* acts on ${\prod}_{r=0}^{\infty}{\otimes}^{4+r}{\mathbb{R}}^{n+2\ast}}.$

(p.91) For

P = pa, we have $\phi (x)={a}^{-1}x$and $\Omega ={a}^{-2}$ By Proposition 6.5, the component ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$

for *a ^{-2}g* is that for

*g*multiplied by ${a}^{2{s}_{\infty}-2}$ Since ${({\phi}^{\prime}{(0)}^{-1})}^{*}$ acts by multiplying this component by ${a}^{{s}_{M}}$ it follows that the

components of *c*(p* _{a}*.g) are those of

*c(g)*multiplied by ${a}^{{s}_{M}+2{s}_{\infty}-2}={a}^{S-2}$ But we noted above that this is precisely how

*p*acts on ${\prod}_{r=0}^{\infty}\left({\otimes}^{4+r}{\mathbb{R}}^{n+2}{}^{*}\otimes {\sigma}_{-2-r}\right)}.$ Finally, for

_{a}*P = pb*, we have

*φ'(0) = I*and $\Omega =1-{b}_{i}{x}^{i}+O(|x{|}^{2}),$ so that the components of $c({({\phi}^{-1})}^{*}({\Omega}^{2}g))$ are given by Proposition 6.5 with ${\Upsilon}_{i}=-{b}_{i}.$ But this is precisely how

*pb*, acts on ${\prod}_{r=0}^{\infty}{\otimes}^{4+r}}{\mathbb{R}}^{n+2}{}^{*}.$

Let us examine more carefully the equivariance of c with respect to the subgroup ${\mathbb{R}}_{+}\subset P$ The component ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$of *c(g)* is a polynomial in the components of the *g( ^{s})* and this equivariance says that when the

*g*(

^{s}) are replaced by

*a*then ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$

^{s}(^{s})*is multiplied by* ${a}^{S-2}$ with $S=\Vert IJKL{M}_{1}\cdots {M}_{r}\Vert $ In particular, ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ can only involve for *g _{(s)} s >≤ S –* 2. This implies that for each

*N ≥*0 (satisying also N ≤

*n*-3 for

*n*even), c induces a P-equivariant polynomial map ${c}^{N}:{\mathcal{N}}_{c}^{N+2}\to {\tilde{\mathcal{R}}}^{N}.$ Clearly these induced maps satisfy the compatibility conditions ${c}^{N-1}{\pi}_{N+2}={\tilde{\pi}}_{N}{c}^{N},$ where ${\pi}_{N}:{\mathcal{N}}_{c}^{N}\to \text{}{\mathcal{N}}_{c}^{N-1}$ and ${\tilde{\pi}}_{N}:{\tilde{\mathcal{R}}}^{N}\to \text{}{\tilde{\mathcal{R}}}^{N-1}$ are the natural projections. The main result of this chapter is the following jet isomorphism theorem.

Theorem 8.10. *Let N ≤ 0* and assume that if N ≤ n-3*is even. Then ${\tilde{R}}^{N}$is a smooth submanifold of ${\tilde{T}}^{N}$ whose tangent space at* 0 *is $T{\tilde{R}}^{N}$and the map* ${C}^{N}:{N}_{c}^{N+2}\to {\tilde{R}}^{N}$*is a P-equivariant polynomial equivalence.*

For *n* odd, it follows that $c:{\mathcal{N}}_{c}\to \text{}\tilde{\mathcal{R}}$ is a P-equivariant bijection since c is the projective limit of the *c ^{N}.*

It will be convenient in the proof of Theorem 8.10 to use Theorem 8.3 to realize *N ^{c}* in terms of curvature tensors on ℝ

_{n}rather than Taylor coefficients of metrics. So we make the following definition. Recall the space

*R*introduced in Definition 8.2.

Definition 8.11.

Define the space ${\mathcal{R}}_{c}\subset \text{}\mathcal{R}$ to be the subset consisting of lists of tensors

(R_{(0)}_{R(1)},…) satisfying in addition to the conditions in Definition 8.2 the following: for each $r\ge 0$

Here Sym refers to the symmetrization over the free indices $jl{m}_{1}\cdots {m}_{r}$ For $N\ge 0$ by ${\mathcal{R}}_{c}^{N}$ we will denote the corresponding set of truncated lists$({R}^{(0)},{R}^{(1)},\dots ,{R}^{(N)})$

For $r,N\ge 0$ we define also the following finite-dimensional vector spaces:

(8.5) and (1), (2) of Definition 8.2 hold}

The bijection$N\to \text{}\mathcal{R}$ asserted by Theorem 8.3 clearly restricts to a bijection ${N}_{c}\to \mathcal{R}{\text{}}_{c}$ whose truncated maps${N}_{c}^{N+2}\to \text{}{\mathcal{R}}_{c}^{N}$ are polynomial equivalences. By composition, we can regard c and the *c ^{N}* as defined on the corresponding

*R*and R

_{c}^{N}

_{c}In the following, we will not have occasion to refer to ${\mathcal{N}}_{c}$ and ${\mathcal{N}}_{c}^{N+2}$ so no confusion should arise from henceforth using thesame symbols c and

*c*for the maps defined on R

_{N}_{c}and ${\mathcal{R}}_{c}^{N}$ We can transfer the action of

*P*on ${\mathcal{N}}_{c}$ to R

_{c}. The element

*p*acts on R

_{a}_{c}by multiplying ${R}^{(r)}$ by ${a}^{r+2};$ this same prescription gives an ℝ

_{+}action on

*T*for

_{N}*N*≥ 0. The ℝ

_{+}-equivariance of ${c}^{N}$ implies that a component ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ of C

^{N}(R) with $\Vert IJKL{M}_{1}\cdots {M}_{r}\Vert =N+4$ can be written as a linear combination of R

^{(N)}plus quadratic and higher terms in the components of the

*R*with $r\le N-2$

_{(r)}Our starting point for the proof of Theorem 8.10 is the following lemma.

Lemma 8.12.

For each N ≥ 0, the subset ${\mathcal{R}}_{c}^{N}\subset {T}^{N}$ is a smooth submanifold whose tangent space at0is$T{\mathcal{R}}_{c}^{N}.$

Proof.

We will show that for each N ≥ 1, there is a polynomial equivalence ${\Phi}^{N}:{\mathcal{T}}^{N}\to {\mathcal{T}}^{N}$ satisfying $d{\Phi}^{N}(0)=I$ and ${\Phi}^{N}({\mathcal{R}}_{c}^{N-1}\times {\sigma}^{N})=\text{}{\mathcal{R}}_{c}^{N}$ Upon iterating this statement and using ${\sigma}^{0}=\text{}{\mathcal{R}}_{c}^{0}=T{\mathcal{R}}_{c}^{0}$ we conclude the existence of a polynomial equivalence : ${\mathcal{T}}^{N}\to \text{}{\mathcal{T}}^{N}$ whose differential at 0 is the identity, and which maps $T{\mathcal{R}}_{c}^{N}\to \text{}{\mathcal{R}}_{c}^{N}$ The desired conclusion follows immediately.

When reformulated in terms of the spaces ${\mathcal{R}}_{c}^{N},$ Lemma 8.6 asserts the existence for each N ≥ 1 of a polynomial map ${\eta}^{N}:{\mathcal{T}}^{N-1}\to {\otimes}^{N+4}{\mathbb{R}}^{n}{}^{*}$ with zero constant and linear term, such that the map ${\Lambda}^{N}:R\to (R,{\eta}^{N}(R))$ sends ${\mathcal{R}}_{c}^{N-1}\to \text{}{\mathcal{R}}_{c}^{N}$. Here

Rdenotes the list constituting an element ofTThere is no loss of generality in assuming that ${\eta}^{N}({\mathcal{T}}^{N-1})\subset {\tau}^{N}$ so that Ʌ^{N-l}._{N}: ${\mathcal{T}}^{N-1}\to \text{}{\mathcal{T}}^{\mathrm{N.}}$ Define ${\Phi}^{N}:{\mathcal{T}}^{N}\to \text{}{\mathcal{T}}^{N}$ by${\Phi}^{N}(R,{R}^{(N)})=(R,{R}^{(N)}+{\eta}^{N}(R))$ It is evident from the form of the relations defining R^{N}_{c}that ${\Phi}^{N}({\mathcal{R}}_{c}^{N-1}\times {\sigma}^{N})=\text{}{\mathcal{R}}_{c}^{N},$ and clearly ${(}^{{\Phi}^{N}}(R,{R}^{(N)})=(R,{R}^{(N)}-{\eta}^{N}(R))$

We remark that the same proof could have been carried out in terms of the spaces of normal form coefficients, and shows that the subset. ${\mathcal{N}}_{c}^{N}\subset \text{}{\mathcal{N}}^{N}$ is a smooth submanifold whose tangent space at 0 is obtained by linearizing the (p.93) equation obtained by writing (8.5) in terms of the normal form coefficients

${g}_{ij,{k}_{1}\cdots {k}_{r}}.$ At this point we do not know that ${\tilde{\mathcal{R}}}^{N}$ is a submanifold of ${\tilde{T}}^{N}$ with tangent space $T{\tilde{\mathcal{R}}}^{N}$but it is clear that the tangent vector at 0 to a smooth curve in ${\mathcal{R}}^{N}$ must lie in $T{\tilde{\mathcal{R}}}^{N}$ So we conclude for the differential of c^{N} at the origin that $d{c}^{N}:T{\mathcal{R}}_{c}^{N}\to T{\tilde{\mathcal{R}}}^{N}$ The differentiation of the action of *Ρ* on ${\mathcal{R}}_{c}^{N}$ gives a linear action of *Ρ* on $T{\mathcal{R}}_{c}{}^{N}$ and $d{c}^{N}:T{\mathcal{R}}_{c}^{N}\to T{\tilde{\mathcal{R}}}^{N}$ is P-equivariant. By R_{+}-equivariance and linearity of $d{c}^{N}$ … , it follows that *dc ^{N}* decomposes as a direct sum of maps ${\sigma}^{M}\to \text{}{\tilde{\sigma}}^{M}$ for 0≤M≤

*N*By the compatibility of the

*c*as

^{N}*Ν*varies, the map ${\sigma}^{M}\to \tilde{\sigma}{\text{}}^{M}$ is independent of the choice of

*Ν > M*, so we may as well denote it as $d{c}^{N}:{\sigma}^{N}\to \text{}{\tilde{\sigma}}^{N}$ The main algebraic fact on which rests the proof of Theorem 8.10 is the following.

Proposition 8.13.

For Ν >0(and Ν < η –3 if η is even), $d{c}^{N}:{\sigma}^{N}\to \text{}{\tilde{\sigma}}^{N}$is an isomorphism.

Proof of Theorem 8.10 using Proposition 8.13.

We prove by induction on

Nthat there is a polynomial equivalence$\begin{array}{l}{\tilde{\Phi}}^{N}:{\tilde{\mathcal{T}}}^{N}\to \text{}{\tilde{\mathcal{T}}}^{N}\end{array}$ satisfying $d{\tilde{\Phi}}^{N}(0)=I$ and ${\tilde{\Phi}}^{N}({\tilde{\mathcal{R}}}^{N-1}\times \text{}{\tilde{\sigma}}^{N})=\text{}{\tilde{\mathcal{R}}}^{N}$ and that ${c}^{N}:{\mathcal{R}}_{c}^{N}\to \text{}{\tilde{\mathcal{R}}}^{N}$ is a polynomial equivalence. Just as in the proof of Lemma 8.12, iterating the first statement provides a polynomial equivalence : $:{\tilde{\mathcal{T}}}^{N}\to \text{}{\tilde{\mathcal{T}}}^{N}$ whose differential at 0 is the identity and which maps $T{\tilde{\mathcal{R}}}^{N}\to \text{}{\tilde{\mathcal{R}}}^{N}$ from which follows the first statement of Theorem 8.10.For

Ν =0, we can take ${\tilde{\Phi}}^{N}$ to be the identity. Since ${\mathcal{R}}_{c}^{0}=T{\mathcal{R}}_{c}^{0}={\sigma}^{0},$ ${\mathcal{R}}^{0}=T{\mathcal{R}}^{0}=\text{}{\tilde{\sigma}}^{0},$ and c° is linear and can be identified with $d{c}^{0}$ the second statement is immediate from Proposition 8.13.Suppose for some

Ν >1 that we have the polynomial equivalence ${\tilde{\Phi}}^{N-1}$and we know that

cis a polynomial equivalence. Recall the polynomial maps ${\eta}^{N}:{\mathcal{T}}^{N-1}\to {\tau}^{N},{\Lambda}^{N}:{\mathcal{T}}^{N-1}\to {\mathcal{T}}^{N}$ and ${\Phi}^{N}:{\mathcal{T}}^{N}\to \text{}{\mathcal{T}}^{N}$ constructed in Lemma 8.6 and Lemma 8.12. By the induction hypothesis that^{N-1}cis a polynomial equivalence, we conclude that there is a polynomial map ${\tilde{\Lambda}}^{N}:{\tilde{\mathcal{T}}}^{N-1}\to \text{}{\tilde{\mathcal{T}}}^{N}$ such that${\tilde{\Lambda}}^{N}({\tilde{\mathcal{R}}}^{N-1})\subset \text{}{\tilde{\mathcal{R}}}^{N}$ and such that the diagram^{N-1}commutes. Using the compatibility of ${c}^{N-1}$ and

cand the form of the map ${\Lambda}^{N},$ one sees that ${\tilde{\Lambda}}^{N}$ can be taken to have the form ${\tilde{\Lambda}}^{N}(\tilde{\mathcal{R}})=(\tilde{\mathcal{R}},{\tilde{\eta}}^{N}(\tilde{\mathcal{R}}))$^{N}where ${\tilde{\eta}}^{N}:{\tilde{\mathcal{T}}}^{N-1}\to \text{}{\tilde{\tau}}^{N}$ has no constant or linear terms. Now define the map ${\tilde{\Phi}}^{N}:{\tilde{\mathcal{T}}}^{N}\to \text{}{\tilde{\mathcal{T}}}^{N}$ by ${\tilde{\Phi}}^{N}(\tilde{\mathcal{R}},{\tilde{\mathcal{R}}}^{(N)})=(\tilde{\mathcal{R}},{\tilde{\mathcal{R}}}^{(N)}+{\tilde{\eta}}^{N}(\tilde{\mathcal{R}}))$ Clearly$d{\tilde{\Phi}}^{N}(0)=I$ (p.94) and ${\tilde{\Phi}}^{N}({\tilde{\mathcal{R}}}^{N-1}\times \text{}{\tilde{\sigma}}^{N})=\text{}{\tilde{\mathcal{R}}}^{N}$ by the form of the relations defining ${\tilde{\mathcal{R}}}^{N}$ It is a straightforward matter to check that the diagram

commutes. By the induction hypothesis and Proposition 8.13, the vertical map on the left is a polynomial equivalence. We conclude that

cis also a polynomial equivalence, completing the induction step.^{N}

Proof of Proposition 8.13.

The proof has two parts. We will first construct an injective map $L:{\tilde{\sigma}}^{N}\to {\sigma}^{N}.$ We will then show that $d{c}^{N}:{\sigma}^{N}\to \text{}{\tilde{\sigma}}^{N}$ is injective. These statements together imply that $\mathrm{dim}({\sigma}^{N})=\mathrm{dim}({\tilde{\sigma}}^{N}).$ from which it then follows that$d{c}^{N}$ is an isomorphsim.

Let $({\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}})\in \text{}{\tilde{\sigma}}^{N}.$ We can consider the components ${\tilde{R}}_{ijkl,{m}_{1}\cdots {m}_{N}}$ in which all the indices lie between 1 and

n.This defines a map $L:{\tilde{\sigma}}^{N}\to {\Lambda}^{2}{\mathbb{R}}^{n}{}^{*}\otimes {\Lambda}^{2}{\mathbb{R}}^{n}{}^{*}\otimes {\odot}^{N}{\mathbb{R}}^{n}{}^{*}$ and clearly everything in the range ofLsatisfies conditions (1) and (2) of Definition 8.2. We claim that everything in the range ofLalso satisfies (8.5), so that $L:{\tilde{\sigma}}^{N}\to {\sigma}^{N}$ Condition (3) of Definition 8.8 implies ${\tilde{h}}^{IK}{\tilde{R}}_{IjKl,{m}_{1}\cdots {m}_{N}}=0$ This can be written as$${\tilde{R}}_{j\infty l0,{m}_{1}\cdots {m}_{N}}+{\tilde{R}}_{l\infty j0,{m}_{1}\cdots {m}_{N}}+{h}^{ik}{\tilde{R}}_{ijkl,{m}_{1}\cdots {m}_{N}}=0.$$If we apply condition (5) of Definition 8.8 to ${\tilde{R}}_{j\infty l0,{m}_{1}\cdots {m}_{N}}$ and then symmetrize over

jlm_{1}...m_{N}, the result is 0 by the skew symmetry of ${\tilde{R}}^{(N)}$ in the second pair of indices. Similarly for ${\tilde{R}}_{l\infty j0,{m}_{1}\cdots {m}_{N}}$ It follows that the symmetrization of vanishes. This proves that (8.5) holds.Next we show that

Lis injective. We claim that for$({\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}})\in \text{}{\tilde{\sigma}}^{N}$ any component ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ can be written as a linear combination of components in which none of the indicesIJKLMis ∞. We first show that any component can be written as a linear combination of components in which none of_{1}... M_{r}IJKLis ∞. To see this, note that (8.4) and${\tilde{h}}^{AB}{\tilde{R}}_{IJKA,B{M}_{1}\cdots {M}_{r}}=0$ imply

$$-(r+2){\tilde{R}}_{IJK\infty ,{M}_{1}\cdots {M}_{r}}={\tilde{R}}_{IJK\infty ,0{M}_{1}\cdots {M}_{r}}=-{\tilde{R}}_{IJK0,\infty {M}_{1}\cdots {M}_{r}}-{h}^{ab}{\tilde{R}}_{IJKa,b{M}_{1}\cdots {M}_{r}}.$$Thus a component ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ in which $L=\infty $ can be rewritten as a linear combination of components in which $L\ne \infty $ and

UKremain unchanged. Repeating this procedure allows the removal of any ∞'s inIJKL.(p.95) The same method allows the removal of ∞'s inM\ ... M${\tilde{h}}^{AB}{\tilde{R}}_{IJKL,AB{M}_{2}\cdots {M}_{r}}=0$ one has$$-2(r+2){\tilde{R}}_{IJKL,\infty {M}_{2}\cdots {M}_{r}}=2{\tilde{R}}_{IJKL,0\infty {M}_{2}\cdots {M}_{r}}=-{h}^{ab}{\tilde{R}}_{IJKL,ab{M}_{2}\cdots {M}_{r}}.$$Thus all oo's can be removed as indices. Now (8.4) and (5) of Definition 8.8 can be used to remove any O's as indices at the expense of permuting the remaining indices between 1 and

n.It follows that any component ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ can be written as a linear combination of components in which all indices are between 1 andn.ThusLis injective.It remains to prove that $d{c}^{N}$ is injective. If ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}\in {\sigma}^{N}$ we set ${R}_{jl,{m}_{1}\cdots {m}_{N}}={h}^{ik}{R}_{ijkl,{m}_{1}\cdots {m}_{N}},\text{}{R}_{{m}_{1}\cdots {m}_{N}}={h}^{jl}{R}_{jl,{m}_{1}\cdots {m}_{N}}$ and

$$(n-2){P}_{jk,{m}_{1}\cdots {m}_{N}}={R}_{jk,{m}_{1}\cdots {m}_{N}}-{R}_{{m}_{1}\cdots {m}_{N}}{h}_{jk}/2(n-1).$$We also denote by ${W}_{ijkl,{m}_{1}\cdots {m}_{N}}\in {\Lambda}^{2}{\mathbb{R}}^{n}{}^{*}\otimes {\Lambda}^{2}{\mathbb{R}}^{n}{}^{*}\otimes {\odot}^{N}{\mathbb{R}}^{n}{}^{*}$ the tensor obtained by taking the trace-free part in

ijklwhile ignoringm:_{1}... m_{n}$${W}_{ijkl,{m}_{1}\cdots {m}_{N}}={R}_{ijkl,{m}_{1}\cdots {m}_{N}}-2{h}_{l[j}{P}_{i]k,{m}_{1}\cdots {m}_{N}}{h}_{jl}+2{h}_{k[j}{P}_{i]l,{m}_{1}\cdots {m}_{N}}$$Then Wjjfc;

_{imi}…_{mjv}satisfies (1) of Definition 8.2 but not necessarily (2). We also define$${C}_{jkl,{m}_{1}\cdots {m}_{N-1}}=2{P}_{j[k,l]{m}_{1}\cdots {m}_{N-1}}.$$Contracting the second Bianchi identity (2) of Definition 8.2 in the usual way shows that ${W}^{i}{}_{jkl,i{m}_{1}\cdots {m}_{N-1}}=(3-n){C}_{jkl,{m}_{1}\cdots {m}_{N-1}}$ and ${P}^{i}{}_{j,i{m}_{1}\cdots {m}_{N-1}}={P}^{i}{}_{i,j{m}_{1}\cdots {m}_{N-1}}.$

Lemma 8.14.

Let${R}_{ijkl,{m}_{1}\cdots {m}_{N}}\in {\sigma}^{N}$Ifn≥ 4 and ${W}_{ijkl,{m}_{1}\cdots {m}_{N}}=0$ then ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}=0$ Ifn= 3 and ${C}_{jkl,{m}_{1}\cdots {m}_{N-1}}=0$then${R}_{ijkl,{m}_{1}\cdots {m}_{N}}=0$

Proof.IfΝ =0, this follows from the decomposition of the curvature tensor into its Weyl piece and its Ricci piece. Suppose N ≥ 1. Ifn≥ 4, the contracted Bianchi identity above shows that ${C}_{jkl,{m}_{1}\cdots {m}_{N-1}}=0$ which is ourhypothesis ifn= 3. (The hypothesis W_{ijkl,m1...mn}= 0 forn≥ 4 is of course automatic forη =3.) Thus we conclude for anyηthatP= 0, so ${P}_{ij,{m}_{1}\cdots {m}_{N}}={P}_{(ij,{m}_{1}\cdots {m}_{N})}$ Since ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}\in {\sigma}^{N}$ we also have_{j[k,l]m1...mn-1}

R0. Therefore_{(ij,m1...mN)}=$$(n-2){P}_{ij,{m}_{1}\cdots {m}_{N}}=(n-2){P}_{(ij,{m}_{1}\cdots {m}_{N})}=-{R}_{({m}_{1}\cdots {m}_{N}}{h}_{ij)}/2(n-1).$$Now ${h}^{ij}{P}_{(ij,{m}_{1}\cdots {m}_{N})}={h}^{ij}{P}_{ij,{m}_{1}\cdots {m}_{N}}={R}_{{m}_{1}\cdots {m}_{N}}/2(n-1)$ Hence the symmetric tensor $P={P}_{(ij,{m}_{1}\cdots {m}_{N})}$ is in the kernel of the operator

Ρ—» (η –2)Ρ+ Sym(tr(P)/i). This operator is injective on symmetric tensors, so we conclude thatPijtmi---mN =0· The conclusion now follows from (8.6). D(p.96) We will prove that $d{c}^{N}$ is injective by showing that if ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}\in $

ker(dc, then the hypotheses of Lemma 8.14 hold. To get the flavor of the argument, consider first the cases^{N})Ν = 0,1.Now ${\sigma}^{0}$ is the space of trace-free curvature tensors. In Chapter 6, we found that is the Weyl piece of such a curvature tensor. So c^{o}linear and is obviously injective. In Chapter 6, we also calculated the curvature components${\tilde{R}}_{\infty jkl}={C}_{jkl}$ and (see (6.3)). We see that c^{1}is also linear, so can be identified withdcIf ${R}_{ijkl,m}\in \mathrm{ker}(d{c}^{1})$ we first conclude by considering ${\tilde{R}}_{\infty jkl}$ that ${C}_{jkl}=0$ and then by considering ${\tilde{R}}_{ijkl,m}$ that ${W}_{ijkl,m}=0$(for^{1}1.n >4), as desired.For the general case we need to understand the relation between covariant derivatives with respect to the ambient metric and covariant derivatives with respect to a representative

gonM.Recall that the conformai curvature tensors are tensors onMdefined by evaluating components of ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ atρ =0 andt =1. We can take further covariant derivatives of such a tensor with respect tog.We will denote by the tensor onMobtained by such further covariant differentiations. For example, ${\tilde{R}}_{ijkl,|m}={W}_{ijkl,m}$whereas ${\tilde{R}}_{ijkl,m}$ is the tensor ${V}_{ijklm}$ given by (6.3). An inspection of (3.16) shows the relation between ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}p}$ and ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}|p}$ Recalling that ${{g}^{\prime}}_{ij}=2{P}_{ij}$ $\rho =0$at one sees that ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}p}-{\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}|p}$ is a linear combination of components of

V, possibly multiplied by components of_{r}Rg, plus quadratic terms in curvature. Iterating, it follows that ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}{p}_{1}\cdots {p}_{s}}-{\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}|{p}_{1}\cdots {p}_{s}}$ is a linear combination of terms of the form ${\tilde{R}}_{ABCD,{F}_{1}\cdots {F}_{u}|{q}_{1}\mathrm{...}{q}_{t}}$ witht < s, possibly multiplied by components ofg, plus nonlinear terms in curvature.Consider now the map $d{c}^{N}:{\sigma}^{N}\to \text{}{\tilde{\sigma}}^{N}$ The symbol ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}}$ with $\Vert IJKL{M}_{1}\cdots {M}_{r}\Vert =N+4$ is now to be interpreted as the linear function of the ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}$ obtained by applying $d{c}^{N}$ Similarly, we now interpret the symbol ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}|{p}_{1}\cdots {p}_{s}}$ for $\Vert IJKL{M}_{1}\cdots {M}_{r}\Vert +s=N+4$ as a linear function of the ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}$ Suppose that ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}\in \mathrm{ker}(d{c}^{N})$ We claim that ${\tilde{R}}_{IJKL,{M}_{1}\cdots {M}_{r}|{p}_{1}\cdots {p}_{s}}=0$ for $\Vert IJKL{M}_{1}\cdots {M}_{r}\Vert +s=N+4$ The proof is by induction on

s.Fors =0, this is just the hypothesis that ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}\in \mathrm{ker}(d{c}^{N})$ Since we are considering the linearizationdc, the quadratic terms may be ignored in the relation derived in the previous paragraph between ambient covariant derivatives and covariant derivatives on^{N}M.A moment's thought shows that this relation provides the induction step to increase s by 1.Taking

r =0, we conclude that ${\tilde{R}}_{IJKL,|{m}_{1}\cdots {m}_{s}}=0$ For $\Vert IJKL\Vert +s=N+4$ ForIJKL = ijklwe obtain ${W}_{ijkl,{m}_{1}\cdots {m}_{N}}=0$ and forIJKL= ∞jklwe obtain ${C}_{jkl,{m}_{1}\cdots {m}_{N-1}}=0$ Lemma 8.14 then shows that . ${R}_{ijkl,{m}_{1}\cdots {m}_{N}}=0$ as desired.