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.
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 hij of signature (p, q) on . In the positive definite case one typically chooses hij = δij.The background coordinates are geodesic normal coordinates for a metric gij defined near the origin in if and only if gij (0)= hij and the radial vector field ∂r satisfies which is equivalent to or
It is easily seen that the coordinates are normal if and only if
In fact, set and observe that
(p.83) Clearly Fi = 0 implies that (8.1) holds so that the coordinates are normal. Conversely, if the coordinates are normal, then the fact that g and ∂r are parallel along radial lines implies Substituting this and (8.3) into (8.1) gives , which together with the fact that Fi is smooth and vanishes at the origin implies Fi = 0.
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 gij (0)= hij and the derivatives of gij satisfy Since a 3-tensor symmetric in 2 indices and skew in 2 indices must vanish, this implies in particular that all first derivatives of g vanish at the origin. So we define the space of jets of metrics in geodesic normal coordinates as follows.
The space is the set of lists (g(2),g(3), …), where for each satisfies Here the comma just serves to separate the first two indices. For will denote the set of truncated lists (g(2),g(3),…, g(N) ) with the same conditions on the g(r).
To a metric in geodesic normal coordinates near the origin, we associate the element of given by for r ≥ 2. Conversely, to an element of we associate the metric determined to infinite order by these relations together with gij (0)= hij and ∂k gij (0)= 0. In the following, we typically identify the element of and the jet of the metric g.
The space is the set of lists (R(0), R(1),…), where for each and the usual identities satisfied by covariant derivatives of curvature hold:
Here denotes the quadratic polynomial in the 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 to h the metric which contracts the two factors in each term.
For 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 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 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.
The map is bijective and for each Ν ≥0 the truncated map 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 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 .
We now consider the further freedom allowed by the possibility of making conformal changes to g.
Let g be a metric on a manifold M and let p ∈ M. Given with there is uniquely determined to infinite order at p, such that Ω − Ω0 vanishes to second order at p and for all r ≥ 0. Here denotes the full symmetrization of the rank r + 2 tensor
Write and set We are given and . Recall the transformation law for conformal change of Ricci curvature:
Differentiating and conformally transforming the covariant derivative results in
where lots denotes terms involving at most r + 1 derivatives of . We may replace covariant derivatives of by coordinate derivatives on the right-hand side of this formula. Then symmetrizing gives
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 Pij vanish at p, where Pij is given by (3.7). Another is that in normal coordinates, det gij − 1 vanishes to infinite order at p (see [LP]).
The space of jets of metrics in conformal normal form is the subset of consisting of jets of metrics in geodesic normal coordinates for which for all r ≥ 0. For Ν ≥ 2, will denote the subset of 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.
For each Ν ≥ 3, there is a polynomial map with zero constant and linear terms, such that the map maps Here
View g as the metric on a neighborhood of given by the prescribed finite Taylor expansion of order Ν – 1. Then the components of are polynomials in the g(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 + pΝ with pΝ a homogeneous polynomial of degree Ν whose coefficients are polynomials in the g(r) with no constant or linear terms, such that Now Ω2g is in geodesic normal coordinates to order Ν – 1 but not necessarily N. However by the construction of geodesic normal coordinates, there is a diffeomorphism ψ = I + qN+1, where qN+1 is a vector-valued homogeneous polynomial of order N+l whose coefficients are linear in the order Ν Taylor coefficient of Ω2g, such that is in geodesic normal coordinates to order N. Since the condition is invariant under diffeomorphisms, ηΝ defined to be the order Ν Taylor coefficient of has the required properties.
If g1 and g2 are jets of metrics of signature (p, q) at the origin in , we say that g1 and g2 are equivalent if there is a local diffeomorphism ψ defined near 0 satisfying ψ(0) = 0,and a positive smooth function Ω defined near 0, so that 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 Nc. In choosing Ω we have the freedom of and in choosing ψ a freedom of O(p,q). We next describe how these freedoms can be realized as an action on Nc of a subgroup of the conformai group.
Recall that hij is our fixed background quadratic form of signature (p,q) on . Define a quadratic form on ℝn+2 by
and the quadric The metric on ℝn+2 induces a conformal structure of signature (p,q) on Q. The standard action of the orthogonal group on ℝn+2 induces an action on Q by conformal transformations and the adjoint group can be identified with the conformal group of all conformal transformations of Q.
Let and let Ρ be the image in of the isotropy group It is clear that each element of Ρ is represented by exactly one A for which a > 0, so we make the identification The first column of is combining this with the fact that one finds that
where hij is used to raise and lower lowercase indices. It is evident that Ρ = ℝ+ · ℝn · Ο(h), where the subgroups ℝ+, ℝn, Ο (h) arise by varying a, bj, mij, resp.
The intersection of with the cell can be identified with ℝn via where In this identification, the conformal structure is represented by the metric on ℝn. If p ∈ Ρ is as above, the conformal transformation determined by p is realized as
and one has for
This motivates the following definition of an action of Ρ on . Given (p.87) as above and by Proposition 8.4 there is a positive smooth function Ω uniquely determined to infinite order at 0 so that Ω agrees with Ωp to second order and such that for all r > 0. Now so by the construction of geodesic normal coordinates, there is a diffeomorphism φ, uniquely determined to infinite order at 0, so that and such that is in geodesic normal coordinates to infinite order. We define . It is clear by construction of φ that and since the condition of vanishing of the symmetrized covariant derivatives of Ricci curvature is diffeomorphism-invariant, it follows by construction of Ω that It is straightforward to check that this defines a left action of Ρ on . Note that if g = h, then and so that ft is a fixed point of the action. A moment's thought shows that (p.g)(r) depends only on g(s) for s < r. Therefore for each there is an induced action on
It is clear from the construction of the action that p.g is equivalent to g for all and In fact, the P-orbits are exactly the equivalence classes.
The orbits of the Ρ-action on are precisely the equivalence classes of jets of metrics in under diffeomorphism and conformai change.
Proof. It remains to show that equivalent jets of metrics in in the same P-orbit. Suppose that are equivalent. Then we can write to infinite order for a diffeomorphism φ with and a positive smooth Ω. We can uniquely choose the parameters a and b of so that vanishes to second order. Since g1 and g2 both equal h at 0, it follows that so that we can write with Together with the already determined parameters a and b, this choice of m uniquely determines a Since all symmetrized covariant derivatives of Ricci curvature of vanish at the origin, so Ω must be the conformai factor determined when constructing the action of ρ on g1. And since g1 is in geodesic normal coordinates, φ must be the correct diffeomorphism, so that g2 =p.g1.
It is straightforward to calculate from the definition the action on of the ℝ+ and O(h) subgroups of P. If we denote by pa the element of P obtained by taking b = 0 and m = I and by pm the element given by a =1 and b = 0, then one finds that paacts by multiplying g(r) by ar and pmacts by transforming each g(r) as an element of where ℝn denotes the standard defining representation of Ο(h).
(p.88) The problem of understanding local invariants of metrics under diffeomorphism and conformal change reduces to understanding this action of Ρ on NC. However, it is very difficult to analyze or even concretely exhibit the action of the ℝn part of Ρ directly from the definition. The ambient curvature tensors enable the reformulation of the action in terms of standard tensor representations of P.
Propositions 6.1 and 6.4 show that the covariant derivatives of curvature of an ambient metric satisfy relations arising from the homogeneity and Ricci-flatness of the metric. These conditions suggest the following definition.
Definition 8.8.The space is the set of lists such that and such that the following relations hold:
Here, as in Definition 8.2, denotes the quadratic polynomial in the components of which one obtains by differentiating the Ricci identity for commuting covariant derivatives, expanding the differentiations using the Leibnitz rule, and then setting equal to the metric which contracts the two factors in each term. Condition (5) in case r = 0 is interpreted as
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 consisting of the set of lists with such that (l)–(3), (5), (6) hold.
We prepare to define truncated spaces for 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 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 and the conditions (l)–(6) refer to Definition 8.8.
If is a component appearing in an element of and r > N, then at least one of the indices IJKLM1 ... Mr must be 0. Therefore by (5) and (6), can be written as a linear combination of
components with r replaced by r – 1. It follows that
are finite-dimensional. Since (l)–(3), (5) and (6) imply that if we have and As for (4), a typical term in is
where M4' and M4" are lists of indices such that is a rearrangement of In order that it must be that There-fore
This implies that if either is greater than or equal to then the other is less than
or equal to 2. The same reasoning applies to all terms in Since (l)–(3), (5) and (6) imply that a component of an vanishes if its
strength is at most 3, it follows that any component of an which occurs in with a nonzero coefficient must have strength strictly lessHence we will regard (4) as a relation involving components with indices of strength at most 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
: (4) of Definition 8.8 holds}.
We will also need the corresponding linearized spaces, in which the term in (4) is replaced by 0. Define vector spaces
Then Note that in the presence of the condition condition (6) of Definition 8.8 becomes
and is a consequence of (2) and (5). Note also that if
For Let denote the character Since Ρ C Ο (h) and Ρ preserves e0 up to scale, it is easily seen that and are invariant subsets of the P-space where
Rn+22 denotes the standard representation of 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 and '
An easy computation shows that the element acts by multiplying a
component of strength S by
We next define our main object of interest. If g is a metric defined in a neighborhood of 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
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 gij(0)=hij,then 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
for n even. Since the conformai curvature tensors are natural polynomial invariants of the metric g, c is a polynomial map.
The map (or if n is even)is equivariant with respect to the P-actions.
Recall that the action of Ρ on Nc is given by where φ and Ω are determined to map g back to conformai normal form given the initial normalizations defined by p; see the discussion preceding Proposition 8.7 above. By naturality of the conformai curvature tensors,
where on the right-hand side is interpreted as the pullback in the indices between 1 and n of each of the tensors in the list, leaving the 0 and ∞ indices alone. And is given by Proposition 6.5. We use these observations to check for each of the generating subgroups O(h), ℝ++ and ℝn of Ρ that where
the P-action on the right-hand side is that on
For p = pm, we have and is obtained from c(g) by transforming covariantly
underΟ(h) the indices
between 1 and n. But this is precisely how pm acts on
for a-2g is that for g multiplied by Since acts by multiplying this component by it follows that the
components of c(pa.g) are those of c(g) multiplied by But we noted above that this is precisely how pa acts on Finally, for P = pb, we have φ'(0) = I and so that the components of are given by Proposition 6.5 with But this is precisely how pb, acts on
Let us examine more carefully the equivariance of c with respect to the subgroup The component 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 as(s) then
is multiplied by with In particular, 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 Clearly these induced maps satisfy the compatibility conditions where and 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-3is even. Then is a smooth submanifold of whose tangent space at 0 is and the map is a P-equivariant polynomial equivalence.
For n odd, it follows that is a P-equivariant bijection since c is the projective limit of the cN.
It will be convenient in the proof of Theorem 8.10 to use Theorem 8.3 to realize Nc in terms of curvature tensors on ℝn rather than Taylor coefficients of metrics. So we make the following definition. Recall the space R introduced in Definition 8.2.
Define the space 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
Here Sym refers to the symmetrization over the free indices For by we will denote the corresponding set of truncated lists
For we define also the following finite-dimensional vector spaces:
(8.5) and (1), (2) of Definition 8.2 hold}
The bijection asserted by Theorem 8.3 clearly restricts to a bijection whose truncated maps are polynomial equivalences. By composition, we can regard c and the cN as defined on the corresponding Rcand RNc In the following, we will not have occasion to refer to and so no confusion should arise from henceforth using thesame symbols c and cN for the maps defined on Rc and We can transfer the action of P on to Rc. The elementpaacts on Rc by multiplying by this same prescription gives an ℝ+ action on TN for N ≥ 0. The ℝ+-equivariance of implies that a component of CN(R) with can be written as a linear combination of R(N) plus quadratic and higher terms in the components of the R(r) with
Our starting point for the proof of Theorem 8.10 is the following lemma.
For each N ≥ 0, the subset is a smooth submanifold whose tangent space at 0 is
We will show that for each N ≥ 1, there is a polynomial equivalence satisfying and Upon iterating this statement and using we conclude the existence of a polynomial equivalence : whose differential at 0 is the identity, and which maps The desired conclusion follows immediately.
When reformulated in terms of the spaces Lemma 8.6 asserts the existence for each N ≥ 1 of a polynomial map with zero constant and linear term, such that the map sends . Here R denotes the list constituting an element of TN-l. There is no loss of generality in assuming that so that ɅN : Define by It is evident from the form of the relations defining RNc that and clearly
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. 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
At this point we do not know that is a submanifold of with tangent space but it is clear that the tangent vector at 0 to a smooth curve in must lie in So we conclude for the differential of cN at the origin that The differentiation of the action of Ρ on gives a linear action of Ρ on and is P-equivariant. By R+-equivariance and linearity of … , it follows that dcN decomposes as a direct sum of maps for 0≤M≤NBy the compatibility of the cN as Ν varies, the map is independent of the choice of Ν > M, so we may as well denote it as The main algebraic fact on which rests the proof of Theorem 8.10 is the following.
For Ν > 0 (and Ν < η –3 if η is even), is an isomorphism.
Proof of Theorem 8.10 using Proposition 8.13.
We prove by induction on N that there is a polynomial equivalence satisfying and and that is a polynomial equivalence. Just as in the proof of Lemma 8.12, iterating the first statement provides a polynomial equivalence : whose differential at 0 is the identity and which maps from which follows the first statement of Theorem 8.10.
For Ν = 0, we can take to be the identity. Since and c° is linear and can be identified with the second statement is immediate from Proposition 8.13.
Suppose for some Ν > 1 that we have the polynomial equivalence
and we know that cN-1 is a polynomial equivalence. Recall the polynomial maps and constructed in Lemma 8.6 and Lemma 8.12. By the induction hypothesis that cN-1 is a polynomial equivalence, we conclude that there is a polynomial map such that and such that the diagram
commutes. Using the compatibility of and cN and the form of the map one sees that can be taken to have the form
commutes. By the induction hypothesis and Proposition 8.13, the vertical map on the left is a polynomial equivalence. We conclude that cN is also a polynomial equivalence, completing the induction step.
Proof of Proposition 8.13.
The proof has two parts. We will first construct an injective map We will then show that is injective. These statements together imply that from which it then follows that is an isomorphsim.
Let We can consider the components in which all the indices lie between 1 and n. This defines a map and clearly everything in the range of L satisfies conditions (1) and (2) of Definition 8.2. We claim that everything in the range of L also satisfies (8.5), so that Condition (3) of Definition 8.8 implies This can be written as
If we apply condition (5) of Definition 8.8 to and then symmetrize over jlm1...mN, the result is 0 by the skew symmetry of in the second pair of indices. Similarly for It follows that the symmetrization of vanishes. This proves that (8.5) holds.
Next we show that L is injective. We claim that for any component can be written as a linear combination of components in which none of the indices IJKLM1 ... M r is ∞. We first show that any component can be written as a linear combination of components in which none of IJKL is ∞. To see this, note that (8.4) and
Thus a component in which can be rewritten as a linear combination of components in which and UK remain unchanged. Repeating this procedure allows the removal of any ∞'s in IJKL. (p.95) The same method allows the removal of ∞'s in M\ ... M one has
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 can be written as a linear combination of components in which all indices are between 1 and n. Thus L is injective.
It remains to prove that is injective. If we set and
We also denote by the tensor obtained by taking the trace-free part in ijkl while ignoring m1 ... mn:
Then Wjjfc;imi…mjv satisfies (1) of Definition 8.2 but not necessarily (2). We also define
Contracting the second Bianchi identity (2) of Definition 8.2 in the usual way shows that and
Lemma 8.14. Let If n ≥ 4 and then If n = 3 and then
Proof. If Ν = 0, this follows from the decomposition of the curvature tensor into its Weyl piece and its Ricci piece. Suppose N ≥ 1. If n ≥ 4, the contracted Bianchi identity above shows that which is ourhypothesis if n = 3. (The hypothesis Wijkl,m1...mn = 0 for n ≥ 4 is of course automatic for η = 3.) Thus we conclude for any η that Pj[k,l]m1...mn-1 = 0, so Since we also have
R(ij,m1...mN) = 0. Therefore
Now Hence the symmetric tensor 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 that Pijtmi---mN = 0· The conclusion now follows from (8.6). D
ker(dcN), then the hypotheses of Lemma 8.14 hold. To get the flavor of the argument, consider first the cases Ν = 0,1. Now is the space of trace-free curvature tensors. In Chapter 6, we found that is the Weyl piece of such a curvature tensor. So co linear and is obviously injective. In Chapter 6, we also calculated the curvature components and (see (6.3)). We see that c1 is also linear, so can be identified with dc11. If we first conclude by considering that and then by considering that (for 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 g on M. Recall that the conformai curvature tensors are tensors on M defined by evaluating components of at ρ = 0 and t = 1. We can take further covariant derivatives of such a tensor with respect to g. We will denote by the tensor on M obtained by such further covariant differentiations. For example, whereas is the tensor given by (6.3). An inspection of (3.16) shows the relation between and Recalling that at one sees that is a linear combination of components of
Vr R, possibly multiplied by components of g, plus quadratic terms in curvature. Iterating, it follows that is a linear combination of terms of the form with t < s, possibly multiplied by components of g, plus nonlinear terms in curvature.
Consider now the map The symbol with is now to be interpreted as the linear function of the obtained by applying Similarly, we now interpret the symbol for as a linear function of the Suppose that We claim that for The proof is by induction on s. For s = 0, this is just the hypothesis that Since we are considering the linearization dcN, the quadratic terms may be ignored in the relation derived in the previous paragraph between ambient covariant derivatives and covariant derivatives on M. A moment's thought shows that this relation provides the induction step to increase s by 1.
Taking r = 0, we conclude that For For IJKL = ijkl we obtain and for IJKL = ∞jkl we obtain Lemma 8.14 then shows that . as desired.