The Decomposition of Global Conformal Invariants: Some Technical Proofs. I
This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ''global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand...
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irk-123456789-1467882019-02-12T01:23:48Z The Decomposition of Global Conformal Invariants: Some Technical Proofs. I Alexakis, S. This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ''global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. 2011 Article The Decomposition of Global Conformal Invariants: Some Technical Proofs. I / S. Alexakis // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53B20; 53A55 DOI:10.3842/SIGMA.2011.019 http://dspace.nbuv.gov.ua/handle/123456789/146788 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ''global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. |
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Alexakis, S. The Decomposition of Global Conformal Invariants: Some Technical Proofs. I Symmetry, Integrability and Geometry: Methods and Applications |
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Alexakis, S. |
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Alexakis, S. |
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The Decomposition of Global Conformal Invariants: Some Technical Proofs. I |
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The Decomposition of Global Conformal Invariants: Some Technical Proofs. I |
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The Decomposition of Global Conformal Invariants: Some Technical Proofs. I |
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The Decomposition of Global Conformal Invariants: Some Technical Proofs. I |
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The Decomposition of Global Conformal Invariants: Some Technical Proofs. I |
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decomposition of global conformal invariants: some technical proofs. i |
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Інститут математики НАН України |
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2011 |
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The Decomposition of Global Conformal Invariants: Some Technical Proofs. I / S. Alexakis // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 26 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT alexakiss thedecompositionofglobalconformalinvariantssometechnicalproofsi AT alexakiss decompositionofglobalconformalinvariantssometechnicalproofsi |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 019, 41 pages
The Decomposition of Global Conformal Invariants:
Some Technical Proofs. I
Spyros ALEXAKIS
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada
E-mail: alexakis@math.toronto.edu
Received April 01, 2010, in final form February 15, 2011; Published online February 26, 2011
doi:10.3842/SIGMA.2011.019
Abstract. This paper forms part of a larger work where we prove a conjecture of Deser
and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are
defined to be conformally invariant integrals of geometric scalars. The conjecture asserts
that the integrand of any such integral can be expressed as a linear combination of a local
conformal invariant, a divergence and of the Chern–Gauss–Bonnet integrand.
Key words: conormal geometry; renormalized volume; global invariants; Deser–Schwimmer
conjecture
2010 Mathematics Subject Classification: 53B20; 53A55
1 Introduction
The present paper complements the monograph [3] and also [1, 2, 4, 5] in proving a conjecture of
Deser and Schwimmer [15] regarding the algebraic structure of “global conformal invariants”1.
Here we provide a proof of certain lemmas which were announced in the second chapter of [3].
The proofs of these claims do not logically depend on material appearing elsewhere in this entire
work.
For the reader’s convenience, we briefly review the Deser–Schwimmer conjecture.
We recall that a global conformal invariant is an integral of a natural scalar-valued function
of Riemannian metrics,
∫
Mn P (g)dVg, which remains invariant under conformal re-scalings of
the underlying metric2. More precisely, P (g) is assumed to be a linear combination, P (g) =∑
l∈L alC
l(g), where each C l(g) is a complete contraction in the form:
contrl
(
∇(m1)R⊗ · · · ⊗ ∇(ms)R
)
. (1.1)
Here each factor ∇(m)R stands for the mth iterated covariant derivative of the curvature ten-
sor R. ∇ is the Levi-Civita connection of the metric g and R is the curvature associated to this
connection. The contractions are taken with respect to the quadratic form gij . In the present
paper, along with [3, 5] we prove:
Theorem 1.1. Assume that P (g) =
∑
l∈L alC
l(g), where each C l(g) is a complete contraction
in the form (1.1), with weight −n. Assume that for every closed Riemannian manifold (Mn, g)
and every φ ∈ C∞(Mn):∫
Mn
P
(
e2φg
)
dVe2φg =
∫
Mn
P (g)dVg.
1A formulation of the conjecture which is closer to the mathematical terminology used here can also be found
in [23].
2See the introduction of [3] for a detailed discussion on scalar Riemannian invariants and on applications of
the Deser–Schwimmer conjecture.
mailto:alexakis@math.toronto.edu
http://dx.doi.org/10.3842/SIGMA.2011.019
2 S. Alexakis
We claim that P (g) can then be expressed in the form:
P (g) = W (g) + diviT
i(g) + (const) · Pfaff(Rijkl). (1.2)
Here W (g) stands for a local conformal invariant of weight −n (meaning that W (e2φg) =
e−nφW (g) for every φ ∈ C∞(Mn)), diviT
i(g) is the divergence of a Riemannian vector field
of weight −n+ 1, Pfaff(Rijkl) is the Pfaffian of the curvature tensor, and (const) is a constant.
Remark. The notion of a “global conformal invariant” is closely related to the algebraic
structure of conformal anomalies3, which were the motivation of Deser and Schwimmer [15] (see
also especially [23]); however the two notions are not identical: very broadly speaking, a confor-
mal anomaly represents the failure of an action that depends on a metric g to be conformally
invariant. A well-studied such anomaly (which directly gives rise to the global conformal inva-
riants we study here) is considered in [23] in the context of the AdS-CFT correspondence; it
is the renormalized volume A of a Poincaré–Einstein metric (Xn+1, h) with conformal infinity
(Mn, [g]), when n is even. In that setting if ġ = 2φg, then Ȧ(g) =
∫
Mn P (g)φdVg. Furthermore
as noted by Henningson–Skenderis [23], the integral
∫
Mn P (g)dVg is a global conformal invariant
in the sense described above. (This fact was noted in [23] on general grounds; Graham in [21]
showed the conformal invariance of
∫
Mn P (g)dVg by explicitly studying the conformal variation
of the renormalized volume.) Thus, the decomposition (1.2) that we derive for P (g) corresponds
to a decomposition of the integrand in the conformal anomaly; this conforms with the stipulation
of [23]4.
We now digress in order to discuss the relation of this entire work (consisting of the present
paper and [3, 4, 5]) with classical and recent work on local invariants in various geometries.
Local invariants and Fefferman’s program for the Bergman and Szegö kernels.
The theory of local invariants of Riemannian structures (and indeed, of more general geometries,
e.g. conformal, projective, or CR) has a long history. The original foundations of this field were
laid in the work of Hermann Weyl and Élie Cartan, see [26, 14]. The task of writing out local
invariants of a given geometry is intimately connected with understanding which polynomials in
a space of tensors with given symmetries remain invariant under the action of a Lie group. In
particular, the problem of writing down all local Riemannian invariants reduces to understanding
the invariants of the orthogonal group.
In more recent times, a major program was initiated by C. Fefferman in [16] aimed at finding
all scalar local invariants in CR geometry. This was motivated by the problem of understanding
the local invariants which appear in the asymptotic expansions of the Bergman and Szegö
kernels of strictly pseudo-convex CR manifolds, in a similar way to which Riemannian invariants
appear in the asymptotic expansion of the heat kernel; the study of the local invariants in the
singularities of these kernels led to important breakthroughs in [8] and more recently by Hirachi
in [24]. This program was later extended to conformal geometry in [17]. Both these geometries
belong to a broader class of structures, the parabolic geometries; these are structures which admit
a principal bundle with structure group a parabolic subgroup P of a semi-simple Lie group G,
and a Cartan connection on that principle bundle (see the introduction in [12]). An important
question in the study of these structures is the problem of constructing all their local invariants,
which can be thought of as the natural, intrinsic scalars of these structures.
In the context of conformal geometry, the first (modern) landmark in understanding local
conformal invariants was the work of Fefferman and Graham in 1985 [17], where they introduced
their ambient metric. This allows one to construct local conformal invariants of any order in
3These are sometimes called Weyl anomalies.
4See also interesting work of Boulanger in [10] on Weyl anomalies which satisfy the Wess–Zumino consistency
conditions.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 3
odd dimensions, and up to order n
2 in even dimensions. A natural question is then whether all
invariants arise via this construction.
The subsequent work of Bailey–Eastwood–Graham [8] proved that this is indeed true in odd
dimensions; in even dimensions, they proved that the result holds when the weight (in absolute
value) is bounded by the dimension. The ambient metric construction in even dimensions was
recently extended by Graham–Hirachi, [22]; this enables them to identify in a satisfactory manner
all local conformal invariants, even when the weight (in absolute value) exceeds the dimension.
An alternative construction of local conformal invariants can be obtained via the tractor
calculus introduced by Bailey–Eastwood–Gover in [7]. This construction bears a strong resem-
blance to the Cartan conformal connection, and to the work of T.Y. Thomas in 1934 [25]. The
tractor calculus has proven to be very universal; tractor bundles have been constructed [12] for
an entire class of parabolic geometries. The relation between the conformal tractor calculus and
the Fefferman–Graham ambient metric has been elucidated in [13].
The present work, while pertaining to the question above (given that it ultimately deals
with the algebraic form of local Riemannian and conformal invariants), nonetheless addresses
a different type of problem. We here consider Riemannian invariants P (g) for which the integral∫
Mn P (g)dVg remains invariant under conformal changes of the underlying metric. We then
seek to understand the algebraic form of the integrand P (g), ultimately proving that it can be
de-composed in the way that Deser and Schwimmer asserted. It is thus not surprising that the
prior work on the construction and understanding of local conformal invariants plays a central
role in this endeavor, both in [3] and in the present paper.
Index theory. Questions similar to the Deser–Schwimmer conjecture arise naturally in index
theory; a good reference for such questions is [9]. For example, in the heat kernel proof of the
index theorem (for Dirac operators) by Atiyah–Bott–Patodi [6], the authors were led to consider
integrals arising in the (integrated) expansion of the heat kernel over Riemannian manifolds of
general Dirac operators, and sought to understand the local structure of the integrand5. In
that setting, however, the fact that one deals with a specific integrand which arises in the heat
kernel expansion plays a key role in the understanding of its local structure. This is true both
of the original proof of Patodi, Atiyah–Bott–Patodi [6] and of their subsequent simplifications
and generalizations by Getzler, Berline–Getzler–Vergne, see [9].
The closest analogous problem to the one considered here is the work of Gilkey and Branson–
Gilkey–Pohjanpelto [19, 11]. In [19], Gilkey considered Riemannian invariants P (g) for which the
integral
∫
Mn P (g)dVg on any given (topological) manifold Mn has a given value, independent of
the metric g. He proved that P (g) must then be equal to a divergence, plus possibly a multiple of
the Chern–Gauss–Bonnet integrand, if the weight of P (g) agrees with the dimension in absolute
value. In [11] the authors considered the problem of Deser–Schwimmer for locally conformaly
flat metrics and derived the same decomposition (for locally conformaly flat metrics) as in [19].
Although these two results can be considered precursors of ours, the methods there are entirely
different from the ones here; it is highly unclear whether the methods of [19, 11] could be applied
to the problem at hand.
We next review the postponed claims that we will be proving here.
In Section 2 we recall from [3] that P (g) is thought of as a linear combination of complete
contractions involving factors of two types: iterated covariant derivatives of the Weyl tensor
and iterated covariant derivatives of the Schouten tensor6. In other words, we write P (g) =
5We note that the geometric setting in [6] is more general than the one in the Deser–Schwimmer conjecture:
In particular one considers vector bundles, equipped with an auxiliary connection, over compact Riemannian
manifolds; the local invariants thus depend both on the curvature of the Riemannian metric and the curvature of
the connection.
6We refer the reader to the introduction of [3] for a definition of these classical tensors. The Weyl tensor is the
trace-free part of the curvature tensor and is conformally invariant, while the Schouten tensor is a trace-adjustment
of the Ricci tensor (and is not conformally invariant).
4 S. Alexakis∑
l∈L alC
l(g), where each C l(g) is in the form (1.3) below. (Recall from the introduction in [3]
that all complete contractions in P (g) are assumed to have weight −n.)
In Section 2 we prove certain special cases of Propositions 2.3.1, 2.3.2 from Chapter 2 in [3]
that were postponed to the present paper; these are contained in the next lemma:
Lemma 1.1. Let P (g) be a Riemannian invariant of weight −n such that the integral
∫
MnP (g)dVg
is a global conformal invariant. Assume that P (g) is in the form P (g) =
∑
l∈L alC
l(g), where
each C l(g) is a complete contraction:
contr
(
∇(m)W ⊗ · · · ⊗ ∇(m′)W ⊗∇(a)P ⊗ · · · ⊗ ∇(a′)P
)
. (1.3)
Let σ > 0 be the minimum number of factors among the complete contractions C l(g), l ∈ L.
Denote by Lσ ⊂ L the corresponding index set. We then claim that if σ ≤ 2, there exists a local
conformal invariant W (g) and a Riemannian vector field T i(g) such that:
P (g) = diviT
i(g) +W (g) +
∑
l∈L′
alC
l(g),
where each C l(g), l ∈ L′ in the r.h.s. is in the form (1.3) and has at least σ+ 1 factors in total.
In Section 3 we prove the remaining claims for Lemmas 2.5.4, 2.5.5 from Chapter 2 in [3].
Recall the setting and claim of Lemma 2.5.4 in [3]. We recall some definitions. Recall
first that given a P (g) for which
∫
Mn P (g)dVg is a global conformal invariant, we let Isg(φ) to
be the sth conformal variation of P (g): Isg(φ) := ds
dts |t=0e
ntφP (e2tφg). Isg(ψ1, . . . , ψs) is obtained
from Isg(φ) via complete polarization:
Isg(ψ1, . . . , ψs) :=
ds
dλ1 · · · dλs
∣∣∣
λ1=0,...,λs=0
Isg(λ1 · ψ1 + · · ·+ λs · ψs).
We recall that since
∫
Mn P (g)dVg is a global conformal invariant,∫
Mn
Isg(ψ1, . . . , ψs)dVg = 0
for all ψ1, . . . , ψs ∈ C∞(Mn) and all metrics g. We also recall that by virtue of the transformation
laws of the curvature tensor and the Levi-Civita connections under conformal transformations,
Isg(ψ1, . . . , ψs) can be expressed as a linear combination of complete contractions in the form:
contr
(
∇f1...fy∇(m1)Rijkl ⊗ · · · ⊗ ∇g1...gp∇(mr)Rijkl
⊗∇y1...yw∇(d1)Ricij ⊗ · · · ⊗ ∇x1...xp∇(dq)Ricij
⊗∇a1...at1∇(u1)ψ1 ⊗ · · · ⊗ ∇c1...cts∇(us)ψs
)
, (1.4)
with the conventions introduced in Chapter 2 in [3]7. In each factor ∇f1...fy∇(m)
r1...rmRijkl, each of
the upper indices f1 , . . . , fy contracts against one of the indices r1 , . . . , l, while no two of the in-
dices r1 , . . . , l contract between themselves. On the other hand, for each factor ∇y1...yt∇(u)
a1...auψh,
each of the upper indices y1 , . . . , yt contracts against one of the indices a1 , . . . , au . Moreover, none
of the indices a1 , . . . , au contract between themselves. For the factors ∇x1...xp∇(u)
t1...tu
Ricij , we im-
pose the condition that each of the upper indices x1 , . . . , xp must contract against one of the lower
indices t1 , . . . , tu , i, j . Moreover, we impose the restriction that none of the indices t1 , . . . , tu , i, j
contract between themselves (this assumption can be made by virtue of the contracted second
Bianchi identity). We recall some important definitions:
7Note in particular that r is the number of factors ∇(m)Rijkl, q is the number of factors ∇(d)Ric, and s is the
number of factors ∇(u)ψh.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 5
Definition 1.1. We recall that for complete contractions in the form (1.4), δ stands for the
number of internal contractions8, where we are also counting the internal contraction (a, a) in
each factor ∇(p)Ricij = ∇(p)Raiaj and the two internal contractions in each factor R = Rabab.
We also recall that for such complete contractions, |∆| stands for the total number of factors ∆ψh
and R in (1.4). We also recall that a complete contraction in the form (1.4) is called “acceptable”
if each factor ψh is differentiated at least twice. Also, partial contractions in he form (1.4) are
called acceptable is each ψh is differentiated at least twice9.
Furthermore, we recall that among the complete contractions contr(· · · ) in Isg(ψ1, . . . , ψs) with
the minimum number σ of factors in total, µ is the minimum number of internal contractions.
We also recall a definition we have introduced for complete contractions of in the form (1.4):
Definition 1.2. Consider any complete contraction Cg(ψ1, . . . , ψa) in the form (1.4) with σ
factors. If Cg(ψ1, . . . , ψs) has q = 0 and δ = µ it will be called a target. If C lg(ψ1, . . . , ψs) has
q = 0 and δ > µ, it will be called a contributor.
If Cg(ψ1, . . . , ψs) has q > 0 and δ > µ we call it 1-cumbersome. We call Cg(ψ1, . . . , ψs)
2-cumbersome if it has q > 0 and δ = µ and the feature that each factor ∇a1...at∇(p)
r1...rpRicij has
t > 0 and the index j is contracting against one of the indices a1 , . . . , at .
Finally, when we say Cg(ψ1, . . . , ψs) is “cumbersome”, we will mean it is either 1-cumbersome
or 2-cumbersome.
Definition 1.3. We denote by
∑
j∈J ajC
j
g(ψ1, . . . , ψs) a generic linear combination of con-
tributors. We denote by
∑
f∈F afC
f
g (ψ1, . . . , ψs) a generic linear combination of cumbersome
complete contractions.
We have then derived in Chapter 2 in [3] that Isg(ψ1, . . . , ψs) can be expressed in the form:
Isg(ψ1, . . . , ψs) =
∑
l∈Lµ
alC
l,ι
g (ψ1, . . . , ψs) +
∑
j∈J
ajC
j
g(ψ1, . . . , ψs)
+
∑
f∈F
afC
f
g (ψ1, . . . , ψs) + (Junk). (1.5)
(We recall from Chapter 2 in [3] that
∑
l∈Lµ alC
l,ι
g (ψ1, . . . , ψs) in the above is a specific linear
combination of targets, which is in precise correspondence with a specific linear combination of
terms in P (g); the precise form of this correspondence is not important for the present paper,
so we do not dwell on this further.)
We also recall from Chapter 2 in [3] that for each q, 1 ≤ q ≤ σ − s, F q ⊂ F stands for the
index set of complete contractions with precisely q factors ∇(p)Ric or R. Recall that for each
complete contraction in the form (1.4) we have denoted by |∆| the number of factors in one of
the forms ∆ψh, R. For each index set F q above, let us denote by F q,∗ ⊂ F q the index set of
complete contractions with |∆| ≥ σ − 2, and F ∗ =
⋃
q>0 F
q,∗.
Important remark. We recall a remark made in Chapter 2 in [3] where we noted that any
Cfg (ψ1, . . . , ψs) with α > 0 factors R (of the scalar curvature) will have δ ≥ µ+ 2α.
The missing claim needed to derive Lemma 2.5.4 in [3] is then the following:
Lemma 1.2. There exists a linear combination of vector fields (indexed in H below), each in
the form (1.4) with σ factors10, so that modulo complete contractions of length > σ:∑
f∈F ∗
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
∑
y∈Y
ayC
y
g (ψ1, . . . , ψs),
8We recall that an “internal contraction” is a pair of indices that belong to the same factor and contract
against each other.
9For future reference, we also recall that a partial contraction with one free index is also called a vector field.
10The vector fields in question are partial contractions in the form (1.4) with one free index and with weight
−n+ 1.
6 S. Alexakis
where the complete contractions indexed in Y are in the form (1.4) with length σ, and satisfy
all the properties of the sublinear combination
∑
f∈F · · · but in addition have |∆| ≤ σ − 3.
The setting and remaining claims for Lemma 2.5.5 in [3]. Recall that in the setting
of Lemma 2.5.5, our point of reference is a linear combination:
Yg(ψ1, . . . , ψs) =
∑
l∈Lµ
alC
l,ι
g (ψ1, . . . , ψs) +
∑
j∈J
ajC
j
g(ψ1, . . . , ψs) + (Junk),
with the same conventions as in the ones under (1.5). The remaining claims of Lemma 2.5.5
(the proof of which was deferred to this paper) are as follows:
Denote by L∗µ ⊂ Lµ, J∗ ⊂ J the index sets of complete contractions in Yg with |∆| ≥ σ − 2,
among the complete contractions indexed in Lµ, J respectively.
Lemma 1.3. We claim that there exists a linear combination of vector fields (indexed in H
below), with length σ, in the form (1.4) without factors ∇(p)Ric, R and with δ = µ so that:∑
l∈L∗µ
alC
l,i1...iµ
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i|i1...iµ
g (ψ1, . . . , ψs)
∇i1υ · · · ∇iµυ
=
∑
l∈L
alC
l,i1...iµ
g (ψ1, . . . , ψs)∇i1υ · · · ∇iµυ,
where the complete contractions indexed in L are in the form (1.4) with no factors ∇(p)Ric or R
and with |∆| ≤ σ − 3.
In the setting L∗µ = ∅, what remains to be shown to complete the proof of Lemma 2.5.5 in [3]
is the following:
Lemma 1.4. Assume that L∗µ = ∅, and J∗ is as above. We then claim that there exists a linear
combination of vector fields (indexed in H below) so that:∑
j∈J∗
ajC
j
g(ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
∑
y∈Y ′
ayC
y
g (ψ1, . . . , ψs),
where the complete contractions indexed in Y ′ are in the form (1.4) with length σ, with no factors
∇(p)Ric or R and have δ ≥ µ+ 1 and in addition satisfy |∆| ≤ σ − 3.
The setting and claims of Lemma 2.5.3 in [3]. Recall that P (g) =
∑
l∈L alC
l(g) is
assumed to be a linear combination of complete contractions in the form (1.3). Recall that σ
is the minimum number of factors (in total) among all complete contractions C l(g); Lσ ⊂ L is
the corresponding index set. Also, s > 0 is the maximum number of factors ∇(a)P among the
complete contractions C l(g) indexed in Lσ; we denote the corresponding index set by Θs ⊂ L′.
We have defined P (g)|Θs :=
∑
l∈Θs
alC
l(g).
Special definition. If s = σ − 2 then P (g)|Θs is “good” if the only complete contraction
in P (g)|Θs with σ − 2 factors P aa is of the form (const) · contr(∆
n
2
−σ−2∇ilWijkl ⊗∇i
′l′Wi′
jk
l′ ⊗
(P aa )σ−2) (when σ < n
2 − 1) or (const) · contr(∇lWijkl⊗∇l′W ijkl′ ⊗ (P aa )σ−2) when σ = n
2 − 1. If
s = σ−1, then P (g)|Θs is “good” if all complete contractions in P (g)|Θs have δW +δP = n
2 −1.11
Lemma 2.5.3 in [3] claims:
11In other words, if there are complete contractions in P (g)|Θs with δW + δP < n
2
− 1 then P (g)|Θs is “good”
if no complete contractions in P (g)|Θs have σ − 2 factors P aa .
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 7
Lemma 1.5. There exists a divergence diviT
i(g) so that
P (g)|Θs − diviT
i(g) =
∑
l∈Θ′s
alC
l(g) +
∑
t∈T
atC
t(g).
Here each Ct(g) is in the form (1.3) and has fewer than s factors ∇(p)P . The complete con-
tractions indexed in Θ′s are in the form (1.3) with s factors ∇(p)P and moreover this linear
combination is “good”.
2 Proof of Lemma 1.1
We first observe that the claim is trivial when σ = 1. In that case (modulo applying a cur-
vature identity and introducing longer correction terms), the sublinear combination P (g)|1 will
be P (g)|1 = (const) · ∆
n
2
−1R (R is the scalar curvature). Thus, P (g)|1 can be written as
(const)∇i(∇i∆
n
2
−2R).
The case σ = 2 is dealt with by explicitly constructing divergences of vector fields and
long calculations of one local conformal invariant, using the Feffeman–Graham ambient metric,
[17, 18]12. We first consider the terms in P (g)|2 with two factors ∇(m)Pab. We will show that
we can explicitly construct a divergence diviT
i(g) so that:
P (g)|2 − diviT
i(g) = (const)|∇(n
2
−2)P |2 +
∑
h∈H
ahC
h(g) +
∑
w∈W
awC
w(g), (2.1)
where the terms indexed in H have one factor ∇(m)Wijkl and one factor ∇(m′)Pab. The terms in-
dexed in W have two factors ∇(m)Wijkl. (The above holds modulo longer complete contractions,
as usual).
We explain how (2.1) is proven in detail, since the main idea will be used repeatedly through-
out this section. Let us first recall a few classical identities, which can be found in [3].
Useful formulae. Firstly, antisymmetrizing the indices c, a in a factor ∇(m+1)
r1...rmcPab,
13 gives
rise to a Weyl tensor:
∇(m+1)
r1...rmcPab −∇
(m+1)
r1...rmaPcb =
1
3− n
∇(m)
r1...rmsWcab
s. (2.2)
We also recall that the indices i, j and k, l in each tensor ∇(m)
r1...rMWijkl are anti-symmetric.
Finally, we recall the “fake” second Bianchi identities from Chapter 2 in [3]. These are substitutes
for the second Bianchi identity for the tensor ∇aWijkl.
Now, consider any complete contraction C(g) involving exactly two factors, ∇(m)Pab and
∇(m′)Pa′b′ . So C(g) = contr(∇(m)Pab ⊗ ∇(m′)Pa′b′). We firstly show that by subtracting an
explicitly constructed divergence14 diviT
i(g) =
∑K
h=1 diviC
K,i(g) we can write:
C(g)− diviT
i(g) = C ′(g) (2.3)
(modulo terms with more than two factors), where C ′(g) is some complete contraction involving
two factors T1 = ∇(n
2
−2)Pab, T2 = ∇(n
2
−2)Pa′b′ with the additional property that each of the n
2
indices in the first factor contract against an index in the second factor, and vice versa.
12We refer the reader to Chapter 2 in [3] for a review of the ambient metric and and of the algorithm we employ
for computations.
13This is the (m+ 1)st iterated covariant derivative of the Schouten tensor, see [3] for details.
14As allowed in the Deser–Schwimmer conjecture.
8 S. Alexakis
Proof of (2.1). We construct the divergences needed for (2.3). Let us suppose that C(g) has M
pairs of indices (s, t) which belong to the same factor and contract against each other; we call
such contractions “internal contractions”. If M = 0 C(g) is in the desired form; thus we may
assume that M > 0. Firstly, by using the second contracted Bianchi identity ∇cP aa = ∇aP ac ,
we may assume that if one of the factors ∇(m)Pab in C(g) has m > 0, then the indices a, b
are not contracting against each other. After this, we construct the divergence by an iterative
procedure. Pick out any pair of indices s, t in C(g) which contract against each other and
belong to the same factor; assume without loss of generality that s is a derivative index15. We
then construct a partial contraction C1,i(g) out of C(g) by erasing the (derivative) index s and
making the index t into a free index i. Observe then that (modulo complete contractions with at
least three factors), C(g)− diviC
1,i(g) = C(g), where C(g) has M − 1 “internal contractions”.
Iterating this step M − 1 more times, we derive (2.3).
Now, consider C ′(g) and consider the indices a, b in the factor T1 = ∇(n
2
−2)Pab. They contract
against two indices (say c, d) in the factor T2. We then apply the curvature identity repeatedly
and also the identity (2.2) to arrange that the indices a, b in T1 contract against the indices c, d
in T2. The correction terms that arise by the application of the curvature identity have three
factors. The correction terms arising from (2.2) will be in the generic form
∑
h∈H ahC
h(g) as
described below (2.1). �
Now, we will show that (const) = 0 in (2.1). We derive this easily. Consider I2
g (φ) (:=
d2
dt2
|t=0[entφP (e2tφg)]) and apply the super divergence formula to the above16. We derive that
(const) · |∇(n
2
)φ|2 = 0 (modulo longer terms). Therefore, (const) = 0.
Thus, we may assume without loss of generality that
P (g)|2 =
∑
h∈H
ahC
h(g) +
∑
w∈W
awC
w(g),
with the same conventions introduced under (2.1).
We next claim that we can explicitly construct a divergence diviT
′i
g such that:∑
h∈H
ahC
h(g)− diviT
′i(g) =
∑
w∈W
awC
w(g),
where the terms indexed in W are of the generic form described above. The divergence needed for
the above is constructed by the same technique as for (2.3): In each Ch(g) we iteratively pick out
the internal contractions in the factor∇(m)Wijkl, erase one (derivative) index in that contraction,
(thus obtaining a partial contraction with one free index) and subtract the divergence of that
partial contraction17. After repeating this process enough times so that no internal contraction
is left in the factor ∇(m)Wijkl, we end up with a formula:∑
h∈H
ahC
h(g)− diviT
′i(g) =
∑
r∈R
arC
r(g).
Here the complete contractions Cr(g) in the r.h.s. are in the form contr(∇(m)Wijkl ⊗∇(t)Pab),
where all m+ 4 indices in the factor ∇(m)Wijkl contract against the other factor ∇(t)Pab. Now,
15This assumption can be made by virtue of the previous sentence and since we are dealing with complete
contractions of weight −n.
16See the algorithm at the end of [1].
17Sometimes, by abuse of language, we will refer to this subtraction of an explicit divergence as an “integration
by parts”.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 9
since the indices a, b in Schouten tensor are symmetric and the indices i, j and k, l are antisym-
metric, it follows from (2.2) and from the curvature identity that:∑
r∈R
arC
r(g) =
∑
w∈W
awC
w(g),
modulo complete contractions with three factors, which arise due to the curvature identity.
So we may assume without loss of generality that P (g)|2 consists of terms with two factors
∇(m)Wijkl.
We can then again explicitly construct a divergence diviT
′′i(g), in order to write (modulo
longer complete contractions):
P (g)|2 − diviT
′′i(g) =
∑
w∈W ′
awC
w
g ,
where the terms in the r.h.s. have the additional feature that none of the two factors ∇(m)Wabcd
have internal contractions. The divergence diviT
′′i(g) is constructed by the iterative procedure
used to prove (2.3). So we may assume without loss of generality that all terms in P (g)|2 have
this property.
Finally, by just keeping track of the correction terms of the form ∇sWsdfh ⊗ g in the “fake”
second Bianchi identities18, we will prove that we can can explicitly construct a divergence
diviT
i(g) such that:
P (g)|2 = (const)′C∗(g) + diviT
i(g), (2.4)
where C∗(g) is the complete contraction |∇(n
2
−2)Wabcd|2.
Proof of (2.4). By virtue of the anti-symmetry of the indices i, j and k, l and of the first
Bianchi identity in the Weyl tensor Wijkl, we see that (modulo introducing correction terms
with three factors), P (g)|2 can be expressed in the form:
P (g)|2 = a ·
∣∣∇(n
2
−2)Wabcd
∣∣2 + b · C2(g) + c · C3(g),
where C2(g) is the complete contraction:
contr
(
∇r1...rn
2−3sWtjkl ⊗∇
r1...rn
2−3tW sjkl
)
while C3(g) is the complete contraction:
contr
(
∇r1...rn
2−4suWtjky ⊗∇
r1...rn
2−4tyW sjku
)
.
We then only have to apply the “fake” second Bianchi identities from Chapter 2 in [3] to
derive that we can write:
C2(g) =
1
2
∣∣∇(n
2
−2)Wabcd
∣∣2 +
1
n− 3
∣∣∇(n
2
−3)∇sWsjkl
∣∣2. (2.5)
By virtue of (2.5) we easily derive that we can construct a divergence diviT
i(g) so that:
C2(g)− diviT
i(g) =
n− 3
2(n− 4)
∣∣∇(n
2
−2)Wabcd
∣∣2
(notice the constant is strictly positive)19.
18These are presented in Chapter 2 in [3].
19The divergence is constructed in the same way as in the proof of (2.3). We just “integrate by parts” the two
internal contractions s, s in the two factors in |∇(n
2
−3)∇sWsjkl|2.
10 S. Alexakis
In order to deal with the complete contraction C3(g), we will note another useful identity. Let
C
3
(g) stand for the complete contraction (∇r1...rn
2−4a
lWijkl⊗∇
r1...rn
2−4isW ajk
s); we also denote
by C ′3(g) the complete contraction: (∇r1...rn
2−3sWijk
s ⊗∇r1...rn2−3tW ijk
t). We then calculate:
C
3
(g) =
1
2
C ′3(g). (2.6)
Thus, using the above we derive as before that we can explicitly construct a divergence
diviT
′i(g) such that:
C3(g)− diviT
′i(g) =
1
2
C2(g). (2.7)
Therefore, we derive that modulo subtracting a divergence diviT
i(g) from P (g)|2, we may
assume that P (g)|2 is in the form P (g)|2 = (const) · |∇(n
2
−2)W |2. �
A construction in the Fefferman–Graham ambient metric. The reader is referred
to the discussion on the Fefferman–Graham ambient metric from Subsection 3.2.1 in [3]. We
consider the local conformal invariant: C̃(g) = ∆
n
2
−2
g̃ |R̃|2g̃. Here we think of |R̃|2g̃ as the product
g̃aαg̃bβ g̃cγ g̃dδR̃abcdR̃αβγδ, where R̃ is the ambient curvature tensor, g̃ is the ambient metric tensor
and ∆g̃ is the Laplace–Beltrami operator associated to g̃.
We will show that there exists an explicit divergence diviT
i(g), so that modulo terms of
length ≥ 3:
C̃(g) = (const)C∗(g) + diviT
i(g). (2.8)
(Here the constant (const) is non-zero.) If we can prove the above then in view of (2.4) our
claim will clearly follow.
Proof of (2.8). We recall certain facts about the ambient metric The reader is referred to [18],
or Chapter 2 in [3]. Firstly recall that given any point P ∈ M and coordinates {x1, . . . , xn},
there is a canonical coordinate system {t = x0, x1, . . . , xn, ρ} around the image P̃ of P in the
ambient manifold (G̃, g̃). Recall that the local conformal invariant is evaluated at the point P̃ ,
and t(P̃ ) = 1, ρ(P̃ ) = 0.
Recall also that the vector fields X0, X1, . . . , Xn, X∞ are the coordinate vector fields ∂
∂x0 , . . . ,
∂
∂xn ,
∂
∂ρ . Further, when we give values 0, 1, . . . , n,∞ to indices of tensors that appear further
down, these values correspond to the coordinate frame above.
Now, we recall some basic facts regarding the Taylor expansion of the ambient metric g̃ at P̃ .
Firstly, that ∂∞g̃
ab = −2P ab if 1 ≤ a, b ≤ n and ∂∞g̃
∞∞ = −2. These are the only non-zero
components of the matrix ∂∞g̃
cd (with raised indices).
Now, in order to state our next claim, we recall that Γ̃cab stand for the Christoffel symbols
of the ambient metric. We recall that if 0 ≤ a, b, c ≤ n then ∂∞Γ̃cab = F cab(R), where the
expression in the r.h.s. stands for a tensor (in the indices a, b,
c) involving at least one (possibly
internally contracted) factor of the curvature tensor. On the other hand, we also recall that if
1 ≤ a, b, c ≤ n then Γ̃cab = Γcab (the r.h.s. stands for the Christoffel symbol of the metric g).
Finally, if 1 ≤ a, b ≤ n then Γ̃∞ab = −gab. Furthermore, we will use the fact that ∂∞g̃ab = 2Pab
and also the formula (3.21) from [18]:
∂s∞g̃ij =
2
(4− n) · · · (2s− n)
[
∆s−1Pij −∆s−2∇ijP aa
]
+Q(R)
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 11
(for s ≥ 2; all other components ∂
(s)
∞...∞g̃Ab vanish if A = 0,∞ and s ≥ 2; b = 0, 1, . . . , , n,∞)20.
We calculate that for 1 ≤ i, j, k, l ≤ n and 1 ≤ α ≤ n
2 − 2:
∂α∞R̃ijkl = (−1)α−1 1
(n− 3)(n− 4) · · · (n− 2α)
∆α−1
[
∇tjWtikl −∇tiWtjkl
]
+Q(R) (2.9)
(where if α = 1 then the constant above is 1
n−3). Furthermore for 0 ≤ α ≤ n
2 − 3:
∂α∞R̃∞jkl = (−1)α−1 1
(n− 3)(n− 4) · · · (n− 2− 2α)
∆α∇sWsjkl +Q(R) (2.10)
(where if α = 0 then the constant above is 1
n−3). Moreover for 0 ≤ α ≤ n
2 − 4:
∂α∞R̃∞jk∞ = (−1)α
1
(n− 3)(n− 4) · · · (n− 4− 2α)
∆α∇ilWijkl +Q(R). (2.11)
(The left-hand sides of the above are also known as Graham’s extended obstruction tensors,
see [20] for a detailed study of these tensors.)
Next, two calculations. Let F (g̃) be any Riemannian invariant in the ambient metric g̃, with
weight w. (In particular, F (g̃) will have homogeneity −w in t.) Then we calculate:
∂k∞∆g̃[F (g̃)] = ∆gL(g) + (2w + n− 2k)∂k+1
∞ F (g̃) +Q(R). (2.12)
L(g) is a Riemannian invariant of the metric g, not the ambient metric g̃, with weight w − 2k.
Q(R) is a linear combination of complete contractions in the iterated covariant derivatives of
the curvature tensor, and each complete contraction involves at least two such curvature terms.
A word regarding the derivation of the above: By the form of the ambient metric (see in
particular page 20 in [18]), we derive that at any point on the ambient manifold,
∆g̃F (g̃) =
n∑
i,j=1
g̃ij∇(2)
ij F (g̃) + 2t−1∇̃(2)
0∞F (g̃)− 2t2ρ∇̃(2)
∞∞F (g̃)
=
n∑
i,j=1
g̃ij
(
∂
(2)
ij −
n∑
k=1
Γ̃kij∂k
)
F (g̃)−
n∑
i,j=1
g̃ijΓ̃0
ij∂0 +
n∑
i,j=1
g̃ijΓ̃∞ij ∂∞
F (g̃)
+ 2wt−1∂∞F (g̃)− 2t2ρ∂(2)
∞∞F (g̃).
Thus, if we take the kth derivative ∂
(k)
∞...∞ of the above equation and then evaluate at t = 1,
ρ = 0, we obtain the r.h.s. of (2.12) as follows: the term ∆gL(g) arises when all k derivatives ∂∞
hit the factor F (g̃) in
∑n
i,j=1 g̃
ij∂
(2)
ij F (g̃). The coefficient 2w arises from the term ∂
(2)
0∞F (g̃)
in ∇̃(2)
0∞F (g̃), due to the homogeneity w of F (g̃) in x0 = t. The coefficient +n arises from the
term −
∑n
i,j=1 g̃
ijΓ̃∞ij ∂∞. Finally, the term −2k arises when exactly one of the k derivatives ∂∞
hits the coefficient −2t2ρ of the expression ∂
(2)
∞∞F (g̃). All other terms that arise are in the
form Q(R).
Now, we can iteratively apply the above formula to obtain a useful expression for ∆
n
2
−2
g̃ |R̃ijkl|2g̃;
we consider ∆g̃(∆g̃(. . . (|R̃ijkl|2g̃) . . . ), and we replace each of the ∆g̃’s according to (2.12), from
left to right. The resulting equation is:
C̃(g) = (const)′ · ∂
n
2
−2
∞ (|R̃ijkl|2g̃) + Cubic(R) + ∆gL(g).
20Q(R) stands for a linear combination of partial contractions involving at least two curvature terms, as usual.
12 S. Alexakis
Here the constant (const)′ is non-zero; this is because at each application of the identity (2.12)
we have −w + 2k + 2 = n, and w takes on the values −n + 2, . . . ,−4; thus each factor in the
product is non-zero. Cubic(R) is a linear combination of Riemannian invariants with at least
three factors. L(g) is a Riemannian invariant (of the base metric g) of weight −n+ 2.
Now, using formulas (2.9), (2.10), (2.11) and also the formula ∂∞g̃
∞∞ = −2, we derive that:
∂
n
2
−2
∞
(
|R̃ijkl|2g̃
)
= (−1)
n
2
{
2 · 2 1
(n− 3)(n− 4) · · · 4
Wijkl∆
n
2
−3∇itW tjkl
+ 2
n
2
−3∑
x=1
(n
2 − 2
x
)[
1
(n− 3)(n− 4) · · · (n− 2x)
]
×
[
1
(n− 3)(n− 4) · · · (4 + 2x)
]
∆x∇asWsjkl∆
n
2
−2−x∇atW tjkl +
∑
f∈F
afC
f (g)
+
n
2
−3∑
x=0
2 · 4
(n
2 − 2
1
)
·
(n
2 − 3
x
)
·
[
1
(n− 3)(n− 4) . . . (n− 2x− 2)
]
×
[
1
(n− 3)(n− 4) . . . (4 + 2x)
]
∆x∇sWsjkl∆
n
2
−3−x∇tW tjkl
+
n
2
−4∑
x=0
2 · 2
(n
2 − 2
1
)
2 · 2 ·
(n
2 − 3
1
)
·
(n
2 − 4
x
)
×
[
1
(n− 3)(n− 4) . . . (n− 4− 2x)
]
·
[
1
(n− 3)(n− 4) · (4 + 2x)
]
×∆x∇stWsjtl∆
n
2
−4−x∇s′t′W s′jt′l
}
+ Cubic(R). (2.13)
Here Cubic(R) stands for a generic linear combination of partial contractions with at least three
factors of the form ∇(m)Rabcd. The terms indexed in F are complete contractions in the form:
contr
(
∆q∇βaWabcd ⊗∆q′∇bαWαβ
cd
)
.
We will show below that by subtracting a divergence we can “get rid” of such complete contrac-
tions, modulo introducing correction terms with at least three factors.
Mini-proof of (2.13). This equation follows by an iterated application of the Leibniz rule21.
Each derivative can hit either one of the curvature tensors R̃ijkl(g̃), or one of the metric ten-
sors g̃ab (with raised indices).
Now, we have sums
∑...
x=··· in (2.13), to which we will refer to as the “first”, “second”
and “third” sum. Firstly, we observe that the expression Wijkl∆
n
2
−3∇itW tjkl (along with its
coefficient) arises when all n2 −2 derivatives ∂∞ hit precisely one of the two (ambient) curvature
tensors – we then use the formula (2.9). Secondly, we observe that the first sum arises when
all derivatives ∂∞ are forced to hit either of the two (ambient) curvature tensors, and moreover
each curvature tensor must be hit by at least one derivative. Thirdly, the second sum arises
when exactly one derivative ∂∞ hits one of the metric tensors g̃AB (recall that the only non-zero
components of ∂∞g̃
ab are ∂∞g̃
ab = −2P ab22 and ∂∞g
∞∞ = −2; notice that the first term may
be discarded since it gives rise to terms with at least three factors). Fourthly, the third sum
arises when exactly two derivatives ∂∞ hit metric terms g̃AB (a different term each). �
21Recall that we think of |R̃|2g̃ as the product g̃aαg̃bβ g̃cγ g̃dδR̃abcdR̃αβγδ.
22When 1 ≤ a, b ≤ n.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 13
Using (2.13) in conjunction with the formulas (2.5) and (2.7), we can derive (2.8). We can
explicitly construct a divergence diviT
i(g) so that for every x:
Wijkl∆
n
2
−3∇itW tjkl − diviT
i(g)
= (−1)
n
2∇(n
2
−3)
r1...rn
2−3
∇sWsjkl ⊗
(
∇(n
2
−3)
)r1...rn
2−3∇tW tjkl, (2.14)
∆x∇stWsjtl ⊗∆
n
2
−4−x∇s′t′W s′jt′l − diviT
i(g)
= (−1)
n
2
1
2
∇(n
2
−3)
r1...rn
2−3
∇sWsjkl ⊗
(
∇(n
2
−3)
)r1...rn
2−3∇tW tjkl, (2.15)
∆x∇sWsjkl ⊗∆
n
2
−3−x∇tW tjkl − diviT
i(g)
= (−1)
n
2
−1∇(n
2
−3)
r1...rn
2−3
∇sWsjkl ⊗
(
∇(n
2
−3)
)r1...rn
2−3∇tW tjkl, (2.16)
∆x∇asWsjkl ⊗∆
n
2
−2−x∇atW tjkl − diviT
i(g)
= (−1)
n
2∇(n
2
−3)
r1...rn
2−3
∇sWsjkl ⊗
(
∇(n
2
−3)
)r1...rn
2−3∇tW tjkl, (2.17)∑
f∈F
afC
f (g)− diviT
i(g) = 0. (2.18)
All the above equations hold modulo complete contractions with at least three factors. We ex-
plain how the divergences for the above five equations are constructed. Pick out any Laplacian
∆(= ∇aa),23 appearing in any complete contraction above, and formally erase the upper index a.
The resulting partial contraction is a Riemannian 1-tensor field with one free index a. Now,
consider the divergence of this 1-tensor field and subtract it from the original complete contrac-
tion24. The result (modulo correction terms with three factors that arise due to the curvature
identity) is a new complete contraction with one fewer Laplacian. We iterate this step enough
times, until in the end we obtain a complete contraction with no Laplacians. Observe that up to
applying the curvature identity (and thus introducing correction terms with three factors), the
resulting complete contraction is exactly the one claimed in (2.16), (2.17). In the other cases
some additional divergences need to be subtracted. For (2.14) we consider the upper index ∇i in
the right factor; we erase this index (thus obtaining a Riemannian 1-tensor fields) and subtract
the corresponding divergence. The result, up to applying the curvature identity, is the r.h.s.
of (2.14). In the case of (2.15) we perform additional “integrations by parts” by erasing first the
index s in the first factor, and then the index s′ in the second factor. The resulting complete
contraction (after the curvature identity and formula (2.6) is the r.h.s. of (2.15). Finally, to
derive (2.18) we integrate by parts the index b in the expression
∇(q+q′)
t1...tq′r1...rq
∇βaWabcd ⊗∇(q+q′)t1...tq′r1...rq∇bαWαβcd;
the complete contraction we obtain will contain two differentiated Weyl tensors, one of which
is in the form: ∇b...∇aWabcd. This tensor vanishes, modulo a quadratic expression in curvatures,
by virtue of the curvature identity. This concludes our proof of the above five equations.
Therefore, using the above formulas we derive that:
∂
n
2
−2
∞ (|R̃ijkl|2g̃)− diviT
i(g) = (−1)n
{
2 · 2 1
(n− 3)(n− 4) · · · 4
+ 2
n
2
−3∑
x=1
(n
2 − 2
x
)[
1
(n− 3)(n− 4) · · · (n− 2x)
] [
1
(n− 3)(n− 4) · · · (4 + 2x)
]
23A “Laplacian” here means two derivative indices that belong to the same factor and contract against each
other.
24Further we refer to the subtraction of this explicitly constructed divergence as an “integration by parts”.
14 S. Alexakis
−
n
2
−3∑
x=0
2 · 4
(n
2 − 2
1
)
·
(n
2 − 3
x
)
·
[
1
(n− 3)(n− 4) · · · (n− 2x− 2)
]
×
[
1
(n− 3)(n− 4) · · · (4 + 2x)
]
+
n
2
−4∑
x=0
2
(n
2 − 2
1
)
2 · 2 ·
(n
2 − 3
1
)
·
(n
2 − 4
x
)
×
[
1
(n− 3)(n− 4) · · · (n− 4− 2x)
]
·
[
1
(n− 3)(n− 4) · (4 + 2x)
]}
×∇(n
2
−3)
r1...rn
2−3
∇sWsjkl(∇(n
2
−3))
r1...rn
2−3∇tW tjkl. (2.19)
The above holds modulo terms of length at least 3. Now, if we can show that the constant {· · · }
is strictly positive, we will have shown (2.8).
We limit attention to the terms in the first sum with x = y+ 2, the terms in the second sum
with x = y + 1 and in the third sum with x = y (for some given y, 0 ≤ y ≤ n
2 − 4). We observe
that:
2
(n
2 − 2
y + 2
)[
1
(n− 3)(n− 4) · · · (n− 2(y + 2))
] [
1
(n− 3)(n− 4) · · · (4 + 2(y + 2))
]
− 2 · 4
(n
2 − 2
1
)
·
(n
2 − 3
y + 1
)
·
[
1
(n− 3)(n− 4) · · · (n− 2(y + 1)− 2)
]
×
[
1
(n− 3)(n− 4) · · · (4 + 2(y + 1))
]
+ 2
(n
2 − 2
1
)
2 · 2 ·
(n
2 − 3
1
)
·
(n
2 − 4
y
)[
1
(n− 3)(n− 4) · · · (n− 4− 2y)
]
×
[
1
(n− 3)(n− 4) · (4 + 2y)
]
= 2 ·
(n2 − 2) · · ·
(
n
2 − 3− y
)
y!
[
1
(n− 3)(n− 4) · · · (n− 4− 2y)
]
×
[
1
(n− 3)(n− 4) · · · (4 + 2y)
]{
(6 + 2y)(4 + 2y)
(y + 2)(y + 1)
− 4
4 + 2y
y + 1
+ 4
}
= 0.
Now, the only terms we have not taken into account are the term 2 · 2 2
(n−3)(n−4)···4 , the term
in the first sum with x = 1 and the term in the second sum with x = 0. But we observe that
those three terms add up to a positive number:
4
1
(n− 3)(n− 4) · · · 4
+ 2
(n
2
− 2
)
· 1
n− 3
1
(n− 3)(n− 4) · · · 6
− 2 · 4
(n
2
− 2
) 1
n− 3
2
(n− 3)2(n− 4) · · · 4
= 4
1
(n− 3)(n− 4) · · · 4
> 0.
Hence, using the above two formulas we derive that the constant in (2.19) is strictly positive
and thus we derive our claim. This concludes the proof of Lemma 1.1. �
3 The proof of Lemmas 1.2, 1.3, 1.4, 1.5
The three subsections below correspond to the cases s < σ − 2, s = σ − 2, s = σ − 1. We also
prove Lemma 1.5 in the subsections that deal with the cases s = σ − 2, s = σ − 1.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 15
3.1 The proof of Lemmas 1.2, 1.3, 1.4 when s < σ − 2
Brief discussion. Recall the discussion regarding these lemmas in the introduction. Notice
that since s < σ − 2 we have L∗ = J∗ = ∅ by definition; thus in this case we only have to
show Lemma 1.2. Therefore, we will show that there exists a linear combination of tensor fields,
(indexed in H below) so that:∑
f∈F ∗
afC
f
g (ψ1, . . . , φs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , φs) =
∑
f∈FOK
afC
f
g (ψ1, . . . , φs),
where the tensor fields indexed in FOK have all the properties of the “cumbersome” tensor fields,
but moreover have at most σ − 3 factors in the form R or ∆ψh. That will prove Lemma 1.2 in
this setting.
Rigorous discussion. We observe that the complete contractions in Isg(ψ1, . . . , ψs) that
have |∆| ≥ σ − 2 must be contractions Cf indexed in F σ−s
⋃
F σ−s−1
⋃
F σ−s−2. We index the
complete contractions Cf with |∆| = σ − 1 in the sets F σ−s|∆|=σ−1, F σ−s−1
|∆|=σ−1, F σ−s−2
|∆|=σ−1 and the
ones with |∆| = σ − 2 in the sets F σ−s|∆|=σ−2, F σ−s−1
|∆|=σ−2, F σ−s−2
|∆|=σ−2, respectively. (Recall that the
upper labels stand for the number of factors ∇(p)Ric or R in the contractions indexed in F ; the
lower labels stand for the number of factors R plus the number of factors ∆ψh.) We claim:
Lemma 3.1. There is a linear combination of acceptable vector fields of length σ, say∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), so that in the notation above:∑
f∈Fσ−s|∆|=σ−1
⋃
Fσ−1
|∆|=σ−1
⋃
Fσ−s−2
|∆|=σ−1
⋃
Fσ−s|∆|=σ−2
⋃
Fσ−s−1
|∆|=σ−2
⋃
Fσ−s−2
|∆|=σ−2
afC
f
g (ψ1, . . . , ψs)
− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
∑
f∈F q>0,δ>µ
|∆|≤σ−3
afC
f
g (ψ1, . . . , ψs), (3.1)
where the right hand side stands for a generic linear combination of acceptable complete con-
tractions in the form (1.4) with q > 0 factors ∇(p)Ric or R and with δ > µ and |∆| ≤ σ − 3.
We observe that if we can show the above, we will then clearly have proven the remaining case
for our Lemma 1.2. So, the rest of this subsection is devoted to showing the above Lemma 3.1.
Proof of Lemma 3.1. We distinguish two cases and prove them separately. Either σ < n
2 − 1
or σ = n
2 − 1. We begin with the first case.
Proof of Lemma 3.1 in the case σ < n
2
− 1. The proof consists of three steps, which
are spelled out along the proof.
We observe that each complete contraction in the form (1.4) with length σ < n
2 and weight −n
and |∆| = σ − 1 (i.e. a total of σ − 1 factors ∆ψh or R) must have the one factor that is not of
the form ∆ψh or R being in the form:
∇
a1...an−2(σ−1)
2 ∇a1...an−2(σ−1)
2
ψh, (3.2)
or in the form
∇
a1...an−2(σ−1)
2 ∇a1...an−2(σ−1)
2 −2
Rican−2(σ−1)
2 −1
an−2(σ−1)
2
. (3.3)
Therefore all the complete contractions with |∆| = σ − 1 must have δ ≥ µ+ 4.
16 S. Alexakis
First step in the proof of Lemma 3.1. Let Fσ−1 = F σ−s|∆|=σ−1
⋃
F σ−s−1
|∆|=σ−1
⋃
F σ−s−2
|∆|=σ−1. We
claim that there is a linear combination of vector fields in the form (1.4) with length σ, say∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so that modulo complete contractions of length ≥ σ + 1:∑
f∈Fσ−1
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
=
s∑
u=1
(const)uC
u
g (ψ1, . . . , ψs) +
s∑
w=1
w−1∑
q=1
(const)q,wC
w,q
g (ψ1, . . . , ψs)
+ (const)∗C
∗
g (ψ1, . . . , ψs) +
∑
z∈Z
azC
z
g (ψ1, . . . , ψs), (3.4)
where C∗g (ψ1, . . . , ψs) stands for the complete contraction:
contr
(
∇
a1...an−2(σ−1)
2 −1∇a1...an−2(σ−1)
2 −3
Rican−2(σ−2)
2 −1
an−2(σ−1)
2 −1
⊗∇abRicab ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
whereas each Cug (ψ1, . . . , ψs) stands for the complete contraction:
contr
(
∇
a1...an−2(σ−1)
2 −1∇a1...an−2(σ−1)
2 −3
Rican−2(σ−2)
2 −1
an−2(σ−1)
2 −1
⊗∇ababψu
⊗Rσ−s−1 ⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψu ⊗∆ψs
)
,
and each Cw,qg (ψ1, . . . , ψs) stands for the complete contraction:
contr
(
∇
a1...an−2(σ−1)
2 −1∇a1...an−2(σ−1)
2 −1
ψw ⊗∇ababψq
⊗Rσ−s ⊗∆ψ1 ⊗ · · · ⊗ ∆̂ψq ⊗ · · · ⊗ ˆ∆ψw ⊗ · · · ⊗∆ψs
)
.
Finally,
∑
z∈Z azC
z
g (ψ1, . . . , ψs) stands for a generic linear combination of complete contrac-
tions in the form (1.4) with |∆| ≤ σ − 3 and δ ≥ µ + 2. Moreover, we claim that each of the
complete contractions C∗, Cw,q, Cu above has δ ≥ µ+ 4.
Proof of (3.4). We present the proof of this claim in detail, as the same argument will be
used repeatedly in many other instances. We have already observed that we can write out:
∑
f∈Fσ−1
afC
f
g (ψ1, . . . , ψs) = (const)1C
I
g (ψ1, . . . , ψs) +
s∑
y=1
(const)yC
II,y
g (ψ1, . . . , ψs).
Here CI is the complete contraction with one factor as in (3.3) and σ − s − 1 factors R and s
factors ∆ψ1, . . . ,∆ψs. C
II,y is the complete contraction with one factor as in (3.2) and σ − s
factors R and s− 1 factors ∆ψ1, . . . , ˆ∆ψy, . . .∆ψs. In particular, CI is in the form:
contr
(
∇
a1...an−2(σ−1)
2 ∇a1...an−2(σ−1)
2 −2
Rican−2(σ−1)
2 −1
an−2(σ−1)
2
⊗Rσ−s−1 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
. (3.5)
Now, if we can show that we can find vector fields∑
h∈H1
ahC
h,i
g (ψ1, . . . , ψs),
∑
h∈H2
ahC
h,i
g (ψ1, . . . , ψs),
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 17
so that:
CIg (ψ1, . . . , ψs)− divi
∑
h∈H1
ahC
h,i
g (ψ1, . . . , ψs)
=
s∑
u=1
(const)uC
u
g (ψ1, . . . , ψs) +
s∑
w=1
w−1∑
q=1
(const)q,wC
w,q
g (ψ1, . . . , ψs)
+ (const)∗C
∗
g (ψ1, . . . , ψs) +
∑
z∈Z
azC
z
g (ψ1, . . . , ψs) (3.6)
and
CII,yg (ψ1, . . . , ψs)− divi
∑
h∈H2
ahC
h,i
g (ψ1, . . . , ψs)
=
s∑
u=1
(const)uC
u
g (ψ1, . . . , ψs) +
s∑
w=1
w−1∑
q=1
(const)q,wC
w,q
g (ψ1, . . . , ψs)
+ (const)∗C
∗
g (ψ1, . . . , ψs) +
∑
z∈Z
azC
z
g (ψ1, . . . , ψs), (3.7)
then clearly (3.4) will follow. We first prove (3.6).
Proof of (3.6). We define CI,a1
g (ψ1, . . . , ψs) to stand for the vector field that arises from CI
by erasing the index a1 in the first factor (see (3.5)) and making the index a1 that is then left
into a free index. We then calculate:
CIg (ψ1, . . . , ψs)− diva1C
I,a1
g (ψ1, . . . , ψs)
= −(σ − s− 1)CI,Ag (ψ1, . . . , ψs)−
s∑
h=1
CI,hg (ψ1, . . . , ψs),
where CI,A stands for the complete contraction that arises in diva1C
I,a1
g (ψ1, . . . , ψs) when ∇a1
hits one of the factors R and CI,h stands for the complete contraction that arises when ∇a1 hits
the factor ∆ψh. Given CI,A, we then define CI,A,a1
g (ψ1, . . . , ψs) to stand for the vector field
that arises by erasing the index a1 in its first factor (note it is a derivative index) and making
the index a1 in the factor ∇a1R into a free index. Also, for each h, 1 ≤ h ≤ s, we define CI,h,a1
g
to stand for the vector field that arises from CI,hg (ψ1, . . . , ψs) by erasing the index a1 in its first
factor (note it is a derivative index) and making the index a1 in the factor ∇a1∆ψh into a free
index. We then calculate:[
CIg (ψ1, . . . , ψs)− diva1C
I,a1
g (ψ1, . . . , ψs)
]
− (σ − s− 1)diva1C
I,A,a1
g (ψ1, . . . , ψs)− diva1
s∑
h=1
CI,h,a1
g (ψ1, . . . , ψs)
= (σ − s− 1)C∗g (ψ1, . . . , ψs) +
s∑
u=1
Cug (ψ1, . . . , ψs) +
∑
z∈Z
azC
z
g (ψ1, . . . , ψs). (3.8)
(We are using the notation of (3.4).) (3.8) just follows by the definitions. This proves (3.6).
Proof of (3.7). We denote by CII,y,∗g (ψ1, . . . , ψs) the complete contraction:
contr
(
∆
n−2(σ−1)
2
−1ψy ⊗∆R⊗Rσ−s−1 ⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψy ⊗ · · · ⊗∆ψs
)
.
18 S. Alexakis
Then, by complete analogy with the previous case, we explicitly construct a divergence,
divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) from CII,yg (ψ1, . . . , ψs) so that:
CII,yg (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
=
∑
y′ 6=y
Cy,y
′
g (ψ1, . . . , ψs) + CII,y,∗g (ψ1, . . . , ψs) +
∑
z∈Z
azC
z
g (ψ1, . . . , ψs). (3.9)
Now, clearly CII,y,∗g (ψ1, . . . , ψs) is not in the form Cu or C∗ above. Also, for each of the
complete contractions Cy,y
′
g (ψ1, . . . , ψs), we inquire whether y > y′ or y < y′. In the first case,
we actually have a complete contraction in the form Cq,w that is allowed in the right hand side
of (3.7). So in that case, we keep the complete contraction Cy,y
′
.
In the second case, Cy,y
′
is not a complete contraction in one of the forms on the right hand
side of (3.7). Now, by repeating the same argument as before, it follows that we can explicitly
construct a divergence divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) such that y < y′ and deduce that modulo
complete contractions of length ≥ σ + 1:
Cy,y
′
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
= Cy
′,y
g (ψ1, . . . , ψs) +
∑
z∈Z
azC
z
g (ψ1, . . . , ψs),
CII,y,∗g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
= Cy,∗g (ψ1, . . . , ψs) +
∑
z∈Z
azC
z
g (ψ1, . . . , ψs).
Thus, we have shown (3.7) and therefore (3.4).
A study of the complete contractions with |∆| = σ − 2 in (3.1). Now, we focus
on the complete contractions in the index sets F q, q > 0 in (3.1) that have |∆| = σ − 2. We
have observed that only complete contractions in F σ−s, F σ−s−1 and F σ−s−2 can have |∆| =
σ − 2. We have denoted the respective index sets by F σ−s|∆|=σ−2 ⊂ F σ−s, F σ−s−1
|∆|=σ−2 ⊂ F σ−s−1
and F σ−s−2
|∆|=σ−2 ⊂ F σ−s−2. Clearly, since we are dealing with the case s < σ − 2, each complete
contraction Cf indexed in one of the three sets above must have at least one factor R. Hence
(by the “Important remark” in the introduction) it follows that each complete contraction Cf
above has δ ≥ µ+ 2. Therefore, by the definition of F σ−s|∆|=σ−2, we derive that we can write out:∑
f∈Fσ−s|∆|=σ−2
afC
f
g (ψ1, . . . , ψs) =
∑
f∈Fσ−s|∆|=σ−2,A
afC
f
g (ψ1, . . . , ψs)
+
∑
f∈Fσ−s|∆|=σ−2,B
afC
f
g (ψ1, . . . , ψs) +
∑
f∈Fσ−s|∆|=σ−2,C
afC
f
g (ψ1, . . . , ψs).
Here
∑
f∈Fσ−s|∆|=σ−2,A
afC
f
g (ψ1, . . . , ψs) stands for a linear combination of complete contractions
in the form:
contr
(
∇a1...at∇(p)
r1...rpRicij ⊗∇b1...bs∇(q)
z1...zqRici′j′ ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
, (3.10)
with δ ≥ µ + 2 (we observe that all the indices r1 , . . . , rp , i, j in the first factor that are not
involved in an internal contraction must contract against an index z1 , . . . , zq , i′ , j′ in the second
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 19
factor and vice versa).
∑
f∈Fσ−s|∆|=σ−2,B
afC
f
g (ψ1, . . . , ψs) stands for a generic linear combination
of complete contractions in the form:
contr
(
∇a1...at∇(p)
r1...rpRicij ⊗∇b1...by∇(q)
z1...zqψh ⊗R
σ−s−2
⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψh ⊗ · · · ⊗∆ψs
)
, (3.11)
with δ ≥ µ+2 and where y+q ≥ 2 (but the second factor is not in the form ∆ψh, by definition)
and where as above all the indices r1 , . . . , rp , i, j in the first factor that are not involved in an
internal contraction must contract against an index z1 , . . . , zq in the second factor and vice versa.
Finally,
∑
f∈Fσ−s|∆|=σ−2,C
afC
f
g (ψ1, . . . , ψs) stands for a generic linear combination of complete
contractions in the form:
contr
(
∇a1...at∇(p)
r1...rpψh ⊗∇
b1...by∇(q)
z1...zqψh′ ⊗R
σ−s−2
⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψh ⊗ · · · ⊗ ˆ∆ψh′ ⊗ · · · ⊗∆ψs
)
, (3.12)
with δ ≥ µ+2 and where t+p, y+q ≥ 2 (and neither of the first two factors is of the form ∆ψb)
and all the indices r1 , . . . , rp in the first factor that are not involved in an internal contraction
must contract against an index z1 , . . . , zq in the second factor and vice versa.
Similarly, we see that we can write out:∑
f∈Fσ−s−1
|∆|=σ−2
afC
f
g (ψ1, . . . , ψs) =
∑
f∈Fσ−s−1
|∆|=σ−2,α
afC
f
g (ψ1, . . . , ψs) +
∑
f∈Fσ−s−1
|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs),
where
∑
f∈Fσ−s−1
|∆|=σ−2,α
afC
f
g (ψ1, . . . , ψs) stands for a generic linear combination of complete con-
tractions in the form:
contr
(
∇a1...at∇(m)
r1...rmRijkl ⊗∇
b1...bs∇(q)
z1...zqψh ⊗R
σ−s−1
⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψh ⊗ · · · ⊗∆ψs
)
,
with δ ≥ µ + 2 and where s + q ≥ 2 and where as above we observe that all the indices
r1 , . . . , rm , i, j , k, l in the first factor that are not involved in an internal contraction must contract
against an index z1 , . . . , zq in the second factor and vice versa.
Also,
∑
f∈Fσ−s−1
|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs) stands for a generic linear combination of complete
contractions in the form:
contr
(
∇a1...at∇(m)
r1...rmRijkl ⊗∇
b1...bs∇(q)
z1...zqRici′j′ ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
, (3.13)
with δ ≥ µ + 2 and where as above we observe that all the indices r1 , . . . , rp , i, j , k, l in the
first factor that are not involved in an internal contraction must contract against an index
z1 , . . . , zq , i′ , j′ in the second factor and vice versa.
Finally, it follows that:∑
f∈Fσ−s−2
|∆|=σ−2
afC
f
g (ψ1, . . . , ψs) =
∑
f∈Fσ−s−2
|∆|=σ−2,γ
afC
f
g (ψ1, . . . , ψs),
where
∑
f∈Fσ−s−2
|∆|=σ−2,γ
afC
f
g (ψ1, . . . , ψs) stands for a linear combination of complete contractions
in the form:
contr
(
∇a1...at∇(p)
r1...rpRijkl ⊗∇
b1...bs∇(q)
z1...zqRi′j′k′l′ ⊗R
σ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
(3.14)
20 S. Alexakis
(all the indices r1 , . . . , rp , i, j , k, l in the first factor that are not involved in an internal contraction
must contract against an index z1 , . . . , zq , i′ , j′ , k′ , l′ in the second factor and vice versa).
Second step in the proof of Lemma 3.1. The second step consists of proving the
equations (3.15), (3.16), (3.17), (3.18), (3.19), (3.20), (3.21) below. We firstly claim that
we can subtract a divergence, divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), from the sublinear combination∑
f∈Fσ−s|∆|=σ−2,A
afC
f
g (ψ1, . . . , ψs) so as to obtain an equation, modulo complete contractions of
length ≥ σ + 1:∑
f∈Fσ−s|∆|=σ−2,A
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) (3.15)
= (const)1 · C∗g (ψ1, . . . , ψs) +
∑
f∈Fσ−s−1
|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
where C∗g (ψ1, . . . , ψs) stands (as before) for the complete contraction:
contr
(
∇a1...an
2−4ij∇(n
2
−4)
a1...an
2−4
Ricij ⊗∇i
′j′Rici′j′ ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
while
∑
f∈Fσ−s−1
|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs) again stands for a generic linear combination of complete
contractions in the form (3.13) and
∑
z∈Z′ azC
z
g (ψ1, . . . , ψs) stands for a generic linear combi-
nation of complete contractions in the form (1.4) with length σ, δ ≥ µ+ 1 and |∆| = σ − 3.
Next, consider the linear combination
∑
f∈Fσ−s|∆|=σ−2,B
afC
f
g (ψ1, . . . , ψs). We then claim that
there is a divergence of a vector field, divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), so that, modulo complete
contractions of length ≥ σ + 1:∑
f∈Fσ−s|∆|=σ−2,B
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) (3.16)
=
s∑
y=1
(const)y · Cyg (ψ1, . . . , ψs) +
∑
f∈Fσ−s−1
|∆|=σ−2,α
afC
f
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
where the complete contractions Cyg (ψ1, . . . , ψs) are in the form (3.11) with every index involved
in an internal contraction, and the linear combination
∑
z∈Z′ azC
z
g (ψ1, . . . , ψs) is the same as
in (3.15).
Finally, we consider the linear combination
∑
f∈Fσ−s|∆|=σ−2,C
afC
f
g (ψ1, . . . , ψs). Just by swit-
ching two factors, we assume that h > h′ in (3.12). We then claim that we can construct
a divergence of a vector field, divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), so that, modulo complete contrac-
tions of length ≥ σ + 1:∑
f∈Fσ−s|∆|=σ−2,C
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
=
s∑
h=1
h−1∑
h′=1
(const)h,h′ · Ch,h
′
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.17)
where the complete contractions Ch,h
′
g (ψ1, . . . , ψs) are in the form (3.12) with every index in-
volved in an internal contraction and the linear combination
∑
z∈Z′ azC
z
g (ψ1, . . . , ψs) is the same
as above.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 21
More claims. We now consider the generic linear combinations
∑
f∈Fσ−s−1
|∆|=σ−2,α
afC
f
g (ψ1,. . .,ψs),∑
f∈Fσ−s−1
|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs). In order to state our next claims, we define two complete
contractions: C+
g (ψ1, . . . , ψs) will stand for the complete contraction:
contr
(
∇
a1...an−2σ
2 −1
il
∇a1...an−2σ
2 −1
Rijkl ⊗ Ricjk ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
and C++,h
g (ψ1, . . . , ψs), 1 ≤ h ≤ s will stand for the complete contraction:
contr
(
∇
a1...an−2σ
2 −1
il
∇a1...an−2σ
2 −1
Rijkl ⊗∇jkψh ⊗Rσ−s−1
⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψh ⊗ · · · ⊗∆ψs
)
.
Moreover, we define C]g(ψ1, . . . , ψs) to stand for the complete contraction:
contr
(
∇
a1...an−2σ
2 −3
il
∇a1...an−2σ
2 −3
Rijkl ⊗∇adRajkd ⊗Rσ−s−1 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
.
Next claims. Consider the linear combination
∑
f∈Fσ−s−1
|∆|=σ−2,α
afC
f
g (ψ1, . . . , ψs). We claim
that there is a divergence of a vector field, divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), so that, modulo com-
plete contractions of length ≥ σ + 1:∑
f∈Fσ−s−1
|∆|=σ−2,α
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
=
s∑
y=1
(const)′y · C++,y
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.18)
and furthermore, for each C++,y, we have δ(C++,h) ≥ µ+ 2.
On the other hand, we consider the linear combination
∑
f∈Fσ−s−1
|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs). We
then claim that there is a divergence of a vector field, divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), so that,
modulo complete contractions of length ≥ σ + 1:∑
f∈Fσ−s−1
|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) (3.19)
=
∑
f∈Fσ−s−1
|∆|=σ−2,γ
afC
f
g (ψ1, . . . , ψs) + (const)′′ · C+
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs)
and furthermore for C+, we have δ(C+) ≥ µ + 2. (For the definition of
∑
f∈Fσ−s−1
|∆|=σ−2,γ
· · · ,
see (3.14).)
Finally, we claim that there are divergences of vector fields, divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so
that:
C+
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
= C]g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.20)
22 S. Alexakis
modulo complete contractions of length ≥ σ + 1 and also:
C++,h
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.21)
modulo complete contractions of length ≥ σ + 1.
We will now derive (3.15), (3.16), (3.17), (3.18), (3.19), (3.20), (3.21). Before outlining the
proof, however, we note that once we show the above equations, we can derive that there is
a linear combination of acceptable vector fields,
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), so that:∑
f∈Fσ|∆|=σ−1
⋃
Fσ−1
|∆|=σ−1
⋃
Fσ−2
|∆|=σ−1
⋃
Fσ|∆|=σ−2
⋃
Fσ−1
|∆|=σ−2
⋃
Fσ−2
|∆|=σ−2
afC
f
g (ψ1, . . . , ψs)
− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) = (const)∗ · C∗g (ψ1, . . . , ψs)
+
s∑
y=1
(const)y,∗ · Cy,∗g (ψ1, . . . , ψs) +
s∑
w=1
w−1∑
q=1
(const)w,q · Cw,qg (ψ1, . . . , ψs)
+ (const)]C
]
g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs). (3.22)
Proof of (3.15), (3.16), (3.17), (3.18), (3.19), (3.20), (3.21). The divergences needed
in all the above equations are constructed “by hand”, by consecutively picking derivative indices,
making them into free indices and taking the corresponding divergence. As the proof is essentially
the same for the equations (3.15), (3.16), (3.17), (3.18), (3.19), we will only demonstrate the
first one. Afterwards, we show (3.20) and (3.21).
We consider a complete contraction Cfg (ψ1, . . . , ψs) in the form (3.10). We first show that
modulo introducing complete contractions of length ≥ σ + 1, we can subtract a divergence
divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) from Cfg (ψ1, . . . , ψs) and obtain:
Cfg (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
n
2
−σ−4∑
v=2
(const)f,v · Cv,τg (ψ1, . . . , ψs)
+
∑
f∈Fσ−s|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.23)
where Cv,τg (ψ1, . . . , ψs) stands for a complete contraction in the form (3.10) where and all the in-
dices r1 , . . . , rp , i, j and z1 , . . . , zq , i′ , j′ are contracting against a derivative index (such a complete
contraction has δ ≥ µ+ 2) and also the first factor has t = v.
We observe that once we show the above, we can then repeat the proof of equation (3.9) to
this setting to find a divergence divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so that for each v:
Cv,τg (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
= C∗g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.24)
modulo complete contractions of length ≥ σ + 1.
Then, combining equations (3.23) and (3.24), we will deduce (3.15).
We show (3.23) by an induction. Firstly let us observe that the maximum value that δ can
have for a complete contraction in the form (3.10) is n
2 + (σ − s). If δ = n
2 + (σ − s) then all
indices appearing in (3.10) must be involved in an internal contraction (and thus there is nothing
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 23
to show). Now, let us suppose that Cfg (ψ1, . . . , ψs) has δ = P ≥ µ+ 2,25 P < n
2 + (σ − s). We
then construct a vector field Ch,ig (ψ1, . . . , ψs) so that modulo complete contractions of length
≥ σ + 1:
Cfg (ψ1, . . . , ψs)− diviC
h,i
g (ψ1, . . . , ψs)
= (Cf )′g(ψ1, . . . , ψs) +
∑
f∈Fσ−s|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs). (3.25)
Here (Cf )′g(ψ1, . . . , ψs) is a complete contraction in the form (3.10) with δ = P + 1. Clearly, if
we can show the above then by iterative repetition (3.23) will follow.
The vector field Ch,ig (ψ1, . . . , ψs) needed for (3.25) can be easily constructed. We observe
that since σ < n
2 − 1 and the weight is −n, we have that at least one of the first two factors
in (3.10) has at least four indices. With no loss of generality, we assume it is the first factor,
∇a1...at∇(p)
r1...rpRicij for which t+ p+ 2 ≥ 4. Moreover, since we are assuming δ < n
2 + (σ− s), it
follows that at least one of the indices r1 , . . . , rp , i, j is not contracting against an index a1 , . . . , at .
That index can either be a derivative index (say r1 with no loss of generality) or–if all the indices
r1 , . . . , rp are each contracting against one of the indices a1 , . . . , at – an internal index (say i with
no loss of generality).
In the first case, we define Ch,ig (ψ1, . . . , ψs) to be the vector field obtained from Cf be erasing
the derivative index r1 in the first factor and making the index r1 in the second factor in (3.10)
into a free index. We check that for this vector field, (3.25) indeed holds.
In the second case, we see that since t + p + 2 ≥ 4, we can apply the second Bianchi iden-
tity (modulo introducing a complete contraction that will belong to the linear combination∑
f∈Fσ−s|∆|=σ−2,β
afC
f
g (ψ1, . . . , ψs)) and be reduced to the previous case, where the index r1 is
not contracting against one of the indices a1 , . . . , at . Thus, we have proven (3.15). The equa-
tions (3.16), (3.17), (3.18), (3.19) follow by essentially the same reasoning.
We now show (3.20), (3.21). In both cases, we define Cb,lg (ψ1, . . . , ψs) to stand for the vector
field obtained from C+
g (ψ1, . . . , ψs) and C++,h
g (ψ1, . . . , ψs) respectively, by erasing the derivative
index i in the first factor. Then, for (3.21) we define∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) = Cb,ig (ψ1, . . . , ψs).
Since C++,h has δ ≥ µ+ 2, we observe that if ∇i in diviC
b,i
g (ψ1, . . . , ψs) hits the first factor, we
cancel out the complete contraction C++,h. When ∇i hits the second factor, we get a complete
contraction that is equal to a complete contraction of length σ+1 (this is due to the antisymmetry
of the indices i, j). If it hits one of the other factors, we get a complete contraction that belongs
to the linear combination
∑
z∈Z′ azC
z
g (ψ1, . . . , ψs).
The case of (3.20) is more complicated. We again consider the same vector field Cb,ig (ψ1,
. . . , ψs) as above. We then observe that modulo complete contractions of length ≥ σ + 1:
C+
g (ψ1, . . . , ψs)− diviC
b,i
g (ψ1, . . . , ψs) = C ′g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.26)
where C ′ is the complete contraction:
contr
(
∆
n
2
−1−σ∇lRijkl ⊗∇l
′
Rijkl′ ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
.
25This inequality holds by virtue of the “Important remark” in the introduction, and because of the fact that
s < σ − 2.
24 S. Alexakis
Notice that since σ < n
2 − 1, we have t > 1 derivatives on the first factor. We then apply the
second Bianchi identity and write:
C ′g(ψ1, . . . , ψs) = contr
(
∆
n
2
−1−σ−1∇itlRtjkl ⊗∇l
′
Ri
jk
l′
⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
. (3.27)
We then define Cc,ig (ψ1, . . . , ψs) to stand for the vector field that arises from the right hand
side of the above by erasing the index i in the first factor and making the index i that it
contracted against in the second factor into a free index. We observe that:
C ′g(ψ1, . . . , ψs)− diviC
c,i
g (ψ1, . . . , ψs) = C]g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs). (3.28)
Therefore, combining (3.26), (3.27) and (3.28) we have that the vector field needed for (3.20)
is precisely:∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) = Cb,ig (ψ1, . . . , ψs)− Cc,ig (ψ1, . . . , ψs).
The third step of the proof of Lemma 3.1. In view of equation (3.22), by subtracting
the divergence divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) from Isg(ψ1, . . . , ψs), we have obtained a relation,
modulo complete contractions of length ≥ σ + 1:
Isg(ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
∑
l∈Lµ
alC
l,ι
g (ψ1, . . . , ψs)
+
∑
j∈J
ajC
j
g(ψ1, . . . , ψs) +
σ−s−3∑
q1=1
∑
f∈F q1
afC
f
g (ψ1, . . . , ψs) + (const)∗ · C∗g (ψ1, . . . , ψs)
+
s∑
y=1
(const)y,∗ · Cy,∗g (ψ1, . . . , ψs) +
s∑
w=1
w−1∑
q=1
(const)w,q · Cw,qg (ψ1, . . . , ψs)
+ (const)]C
]
g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs). (3.29)
We next claim that (const)∗, (const)y,∗, (const)w,q, (const)] = 0; if we can prove this, we will
then have proven Lemma 3.1 in this case where s < σ − 2.
We first prove that (const)∗ = 0. We denote the r.h.s. of the above by Zg(ψ1, . . . , ψs). Clearly,∫
Mn Zg(ψ1, . . . , ψs)dVg = 0.
We now apply the “main conclusion” of the super divergence formula to this integral equation
(see [1]), deriving an equation supdiv[Isg(ψ1, . . . , ψs,Ω
σ−s)] = 0.26 We focus on the sublinear
combination supdiv∗[I
s
g(ψ1, . . . , ψs,Ω
σ−s)] of complete contractions with δ = 0, s factors ∇ψh
and σ − s − 2 factors ∇Ω. Since the “main conclusion” of the super divergence formula holds
formally, it follows that:
supdiv∗[Zg(ψ1, . . . , ψs,Ω
σ−s)] = 0, (3.30)
modulo complete contractions of length ≥ σ + 1.
Since each Cf in (3.29) has |∆| ≤ σ − 3, it follows that:
supdiv∗[Zg(ψ1, . . . , ψs,Ω
σ−s)] = (−1)
n
2 (const)∗ · Cxg (ψ1, . . . , ψs),
26Recall that Isg(ψ1, . . . , ψs,Ω
σ−s) stands for the linear combination that arises from Isg(ψ1, . . . , ψs) by formally
replacing each factor ∇(p)
r1...rpRicij 6= R by −∇(p+2)
r1...rpij
Ω and each factor R by −2∆Ω.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 25
where Cxg (ψ1, . . . , ψs) is the expression:
contr
(
∇y1...yσ−s−2w1...ws
(
∇x1...xn
2−σ+2Ω⊗∇x1...xn
2−σ+2
Ω
)
⊗∇y1Ω⊗ · · · ⊗ ∇yσ−s−2Ω⊗∇w1ψ1 ⊗ · · · ⊗ ∇wsψs
)
. (3.31)
Since this complete contraction is clearly not zero, we deduce that (const)∗ = 0.
Showing that the other constants in (3.29) are zero follows essentially the same pattern. We
next show that each (const)y,∗ = 0. In order to do that, we again consider the main conclusion of
the super divergence formula applied to Zg(ψ1, . . . , ψs), supdiv[Zg(ψ1, . . . , ψs,Ω
σ−s)] = 0. We
pick out the sublinear combination supdiv∗∗,y[Zg(ψ1, . . . , ψs,Ω
σ−s)] of complete contractions
with length σ, δ = 0 and s− 1 factors ∇ψh, h = 1, . . . , ŷ, . . . , s and σ − 2− (s− 1) factors ∇Ω.
Since the main conclusion of the super divergence holds formally, we deduce that:
supdiv∗∗,y[I
s
g(ψ1, . . . , ψs,Ω
σ−s)] = 0, (3.32)
modulo complete contractions of length ≥ σ + 1.
Analogously to the previous case, we deduce that:
supdiv∗∗,y[I
s
g(ψ1, . . . , ψs,Ω
σ−s)] = (−1)
n
2 (const)y,∗ · Cxxg (ψ1, . . . , ψs),
where Cxxg (ψ1, . . . , ψs) is the complete contraction:
contr
(
∇y1...yσ−s−1w1...ŵy ...ws
(
∇x1...xn
2−σ+2Ω⊗∇x1...xn
2−σ+2
ψy
)
(3.33)
⊗∇y1Ω⊗ · · · ⊗ ∇yσ−s−1Ω⊗∇w1ψ1 ⊗ · · · ⊗ ˆ∇ψy ⊗ · · · ⊗ ∇wsψs
)
. (3.34)
Therefore, we deduce that each (const)y,∗ = 0.
To prove that (const)w,q = 0, we again consider Zg(ψ1, . . . , ψs,Ω
σ−s),27 for which∫
Mn Zg(ψ1, . . . , ψs,Ω
σ−s) = 0. We apply the super divergence formula to this equation (see [1]),
deriving a new local equation, supdiv[Zg(ψ1, . . . , ψs,Ω
σ−s)] = 0; we pick out the sublinear com-
bination supdiv∗∗∗,(w,q)[I
s
g(ψ1, . . . , ψs,Ω
σ−s)] of complete contractions with length σ, δ = 0 and
s − 2 factors ∇ψh, h = 1, . . . , ŵ, . . . , q̂, s and σ − s factors ∇Ω. Since the super divergence
formula holds formally, we deduce that:
supdiv∗∗∗,(w,q)
[
Isg(ψ1, . . . , ψs,Ω
σ−s)
]
= 0, (3.35)
modulo complete contractions of length ≥ σ + 1.
We now claim that:
supdiv∗∗∗,(w,q)
[
Isg(ψ1, . . . , ψs,Ω
σ−s)
]
= (−1)
n
2 (const)] · Cxxxg (ψ1, . . . , ψs), (3.36)
where Cxxxg (ψ1, . . . , ψs) is the complete contraction:
contr
(
∇y1...yσ−s−2z1...ẑw...ẑq ...zs
(
∇x1...xn
2−σ+2ψw ⊗∇x1...xn
2−σ+2
ψq
)
⊗∇y1Ω⊗ · · · ⊗ ∇yσ−sΩ⊗∇w1ψ1 ⊗ · · · ⊗ ˆ∇ψw ⊗ · · · ⊗ ∇̂ψq ⊗ · · · ⊗ ∇wsψs
)
. (3.37)
In order to see this, we recall that we have already shown that (const)∗, (const)II,y,∗, (const)w,q
= 0, and also each other complete contraction Cf in (3.29) has |∆| ≤ σ−3, hence each complete
contraction of length σ in each Tail[Cf ] can have at most σ − 3 factors ∇Ω or ∇ψh. Therefore,
we deduce that (const)(w,q) = 0.
27See the “Main consequence” of the super divergence formula in Subsection 2.2.3 in [3].
26 S. Alexakis
Finally, to show (const)] = 0, we again consider Zg(ψ1, . . . , ψs,Ω
σ−s−2),28 for which we have∫
Mn Zg(ψ1, . . . , ψs,Ω
σ−s−2)dVg = 0. We apply the super divergence formula to this equation,
deriving a new local equation: supdiv[Isg(ψ1, . . . , ψs,Ω
σ−s−2)] = 0; we pick out the sublinear
combination supdiv∗∗∗∗[Zg(ψ1, . . . , ψs,Ω
σ−s−2)] of complete contractions with length σ, δ = 0
and s factors ∇ψh, and σ−s−2 factors ∇Ω. Since the super divergence formula holds formally,
we deduce that:
supdiv∗∗∗∗
[
Zg(ψ1, . . . , ψs,Ω
σ−s−2)
]
= 0,
modulo complete contractions of length ≥ σ + 1.
Analogously to the previous case, we deduce that:
supdiv∗∗∗∗
[
Isg(ψ1, . . . , ψs,Ω
σ−s−2)
]
= (−1)
n
2 (const)] · Cxxxxg (ψ1, . . . , ψs), (3.38)
where Cxxxxg (ψ1, . . . , ψs) is the complete contraction:
contr
(
∇y1...yσ−s−2z1...zs
(
∇x1...xn
2−σ−2i
′l′
Rijk
l ⊗∇x1...xn
2−σ−2ilRi′
jk
l′
)
⊗∇y1Ω⊗ · · · ⊗ ∇yσ−s−2Ω⊗∇w1ψ1 ⊗ · · · ⊗ ∇wsψs
)
.
Hence (const)] = 0. We have shown Lemma 3.1 in the case σ < n
2 − 1.
Proof of Lemma 3.1 in the case σ = n
2
− 1. The case σ = n
2 −1 follows similarly. In this
case, we can apply the same method of explicitly constructing divergences to show that there is
a linear combination of acceptable vector fields,
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs), so that (3.22) holds,
where in this setting C∗g (ψ1, . . . , ψs) is in the form:
contr
(
∇aRicai ⊗∇bRicib ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
whereas Cy,∗g (ψ1, . . . , ψs) is in the form:
contr
(
∇aRicai ⊗∇bib ψy ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψy ⊗ · · · ⊗∆ψs
)
,
and Cq,wg (ψ1, . . . , ψs) is in the form:
contr
(
∇aia ψq ⊗∇bib ψw ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗ ∆̂ψq ⊗ · · · ⊗ ˆ∆ψw ⊗ · · · ⊗∆ψs
)
,
and finally C]g(ψ1, . . . , ψs) is in the form:
contr
(
∇aRijka ⊗∇bRijkb ⊗Rσ−s−2 ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
.
Therefore, by subtracting the divergence divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) from Isg(ψ1, . . . , ψs),
we again obtain (3.29) (modulo complete contractions of length ≥ σ+1), with the new notational
conventions of this setting.
Hence, if we could show that (const)∗, (const)y,∗, (const)w,q, (const)] = 0, we will then have
proven Lemma 3.1 in the case σ = n
2 − 1.
Now, we show that (const)∗, (const)II,y, (const)w,q = 0. As before, we take supdiv[Zg(ψ1, . . . ,
ψs,Ω
σ−s−2)] and focus on the same sublinear combinations supdiv∗, supdiv∗∗, supdiv∗∗∗ as in
the previous case, and then observe that the same equations (3.30), (3.32), (3.35) also hold in
this case. The complete contractions Cx, Cxx, Cxxx are the same as in equations (3.31), (3.34),
(3.37), where we set σ = n
2 − 1.
Now, in order to show that (const)] = 0, we apply the same method as for the previous case:
We consider and focus on the sublinear combination of complete contractions with length σ,
28See the “Main consequence” of the super divergence formula in Subsection 2.2.3 in [3].
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 27
δ = 0, and s factors ∇ψh, σ − 2 − s factors ∇Ω. We denote that sublinear combination
by supdiv∗∗∗∗[I
s
g(ψ1, . . . , ψs,Ω
σ−s−2)]. By the same arguments as before, we again deduce an
equation (3.38), where Cxxxx here stands for the complete contraction:
contr
(
∇y1...yσ−s−2z1...zs
(
∇l′Rijkl ⊗∇lRijkl′
)
⊗∇y1Ω⊗ · · · ⊗ ∇yσ−s−2Ω⊗∇w1ψ1 ⊗ · · · ⊗ ∇wsψs
)
.
We thus deduce that (const)] = 0 in this case also. �
3.2 The proof of Lemmas 1.2, 1.3, 1.4 and Lemma 1.5 when s = σ − 2
We observe that if σ < n
2 − 1 then µ ≤ n
2 . We distinguish subcases, based on σ and µ. Our
three subcases are (σ < n
2 − 1, µ < n
2 ), (σ < n
2 − 1, µ = n
2 ), and σ = n
2 − 1.
Firstly, we consider the subcase where σ < n
2 − 1 and µ < n
2 .
Proof of Lemma 1.5 when s = σ − 2, σ < n
2
− 1. Consider P (g)|Θσ−2 .29 We pick out the
sublinear combination of complete contractions in the form:
contr
(
∇a1...at∇(m1)Wijkl ⊗∇b1...bs∇(m2)Wi′j′k′l′ ⊗ (P aa )σ−2
)
.
We index those complete contractions in the set Θ∗σ−2. We then easily see (by repeating
the explicit constructions from the previous subsection) that we can subtract a divergence from
P (g)|Θ∗σ−2
so that, modulo complete contractions of length ≥ σ + 1:
P (g)|Θ∗σ−2
− divi
∑
h∈H
ahC
h,i(g) = (const)αC
α(g) +
∑
t∈T
atC
t(g),
where Cα(g) in the complete contraction in the form:
contr
(
∆
n
2
−σ−2∇ilWijkl ⊗∇i
′l′Wi′
jk
l′ ⊗ (P aa )σ−2
)
, (3.39)
while30
∑
t∈T atC
t(g) stands for a linear combination of complete contractions in the form:
contr
(
∇a1...at∇(m1)Wijkl ⊗∇b1...bs∇(m2)Wi′j′k′l′ ⊗∇fPfi ⊗ (P aa )σ−3
)
(with fewer than σ − 2 factors P aa and with δ ≥ µ).
Therefore, from now on we may assume with no loss of generality that the sublinear combi-
nation P (g)|Θ∗σ−2
is precisely (const)αC
α(g). �
Proof of Lemmas 1.2, 1.3, 1.4 when s = σ − 2, σ < n
2
− 1, µ < n
2
. We again observe
that the complete contractions in Isg(ψ1, . . . , ψs) with |∆| ≥ σ − 2 will be indexed in the
sets F 2, F 1 and31 in this case, also in the sets J and L, where for each C l, l ∈ L and each Cj ,
j ∈ J we recall that C l, Cj are in the form (1.4) with no factors ∇(p)Ric.
We now again consider Isg(ψ1, . . . , ψs), written out as a linear combination in the form (1.5).
We recall the definition of the index set F ∗ (see the notation above Lemma 1.2). It follows
that F ∗ = F 2
σ−1
⋃
F 2
σ−2
⋃
F 1
σ−2 (recall that the upper labels count the number of factors ∇(p)Ric
or R and the lower labels count the value of |∆|). Furthermore, since P (g)|Θs is “good”, it follows
that L∗µ = ∅, while
∑
j∈J∗ ajC
j
g(ψ1, . . . , ψs) = (const)Cg(ψ1, . . . , ψs), where Cg(ψ1, . . . , ψs) is the
29Recall that Θσ−2 stands for the index set of complete contractions in P (g) (which are in the form
contr(∇(m)W ⊗ · · · ⊗ ∇(m′)W ⊗∇(a)P ⊗ · · · ⊗ ∇(a′)P ) with length σ and with σ − 2 factors ∇(u)P .
30If σ = n
2
− 1 then Cα(g) is in the form contr(∇lWijkl ⊗∇l′W ijkl′ ⊗ (P aa )σ−2).
31Recall that F q ⊂ F stands for the index set of complete contractions with q factors ∇(p)Ric or R.
28 S. Alexakis
complete contraction contr(∆
n
2
−σ−2∇ilRijkl⊗∇i′l′Ri
′jkl′ ⊗∆ψ1⊗· · ·⊗∆ψs) or contr(∇iRijkl⊗
∇i′Ri
′jkl ⊗∆ψ1 ⊗ · · · ⊗∆ψs).
A study of the sublinear combination
∑
f∈F 2
σ−1
afC
f
g (ψ1, . . . , ψs). As before, we see
that since |∆| = σ − 1, s = σ − 2 for each Cf , f ∈ F 2
σ−1,32 it follows that each Cf , f ∈ F 2
σ−1
must have at least one factor R and will hence have δ(Cf ) ≥ µ+ 2 (by the decomposition of the
Weyl tensor).
As before, it follows that:
∑
f∈F 2
σ−1
afC
f
g (ψ1, . . . , ψs) = (const)@C
@
g (ψ1, . . . , ψs) +
σ−2∑
u=1
(const)uC
u
g (ψ1, . . . , ψs),
where here C@
g is the complete contraction:
contr
(
∆(n
2
−σ)R⊗R⊗∆ψ1 ⊗ · · · ⊗∆ψσ−2
)
,
and Cug (ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−σ+1)ψu ⊗R2 ⊗∆ψ1 ⊗ ˆ∆ψu ⊗ · · · ⊗∆ψσ−2
)
.
As before, we denote by
∑
z∈Z′ azC
z
g (ψ1, . . . , ψs) a generic linear combination of complete
contractions in the form (1.4) with length σ, |∆| ≤ σ − 3 and δ ≥ µ + 1. It follows as in
the previous subsection that we can construct a vector field
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so that,
modulo complete contractions of length ≥ σ + 1:∑
f∈F 2
σ−1
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
= (const)!C
!
g(ψ1, . . . , ψs) +
σ−2∑
u=1
(const)!,uC
!,u
g (ψ1, . . . , ψs)
+
σ−2∑
q=1
q−1∑
w=1
(const)!,(q,w)C
!,(q,w)
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
where C !
g(ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−σ−1)R⊗∆R⊗∆ψ1 ⊗ · · · ⊗∆ψσ−2
)
,
C !,u
g (ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−σ−1)R⊗∆2ψu ⊗R⊗∆ψ1 ⊗ ˆ∆ψu ⊗ · · · ⊗∆ψσ−2
)
,
C
!,(q,w)
g (ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−σ)ψq ⊗R2 ⊗∆2ψw ⊗∆ψ1 ⊗ ˆ∆ψw ⊗ · · · ⊗ ∆̂ψq ⊗ · · · ⊗∆ψσ−2
)
.
This follows by the same explicit constructions as for equation (3.4).
A study of the sublinear combination
∑
f∈F 2
σ−2
afC
f
g (ψ1, . . . , ψs). We claim that∑
f∈F 2
σ−2
afC
f
g (ψ1, . . . , ψs) can be expressed as:∑
f∈F 2
σ−2
afC
f
g (ψ1, . . . , ψs) =
∑
v∈V1
avC
v
g (ψ1, . . . , ψs) +
∑
v∈V2
avC
v
g (ψ1, . . . , ψs)
32Recall that the subscript σ−1 means that |∆| = σ − 1 for the complete contractions indexed in F 2
σ−1.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 29
+
∑
v∈V3
avC
v
g (ψ1, . . . , ψs) + (const)spec · Cspec
g (ψ1, . . . , ψs), (3.40)
where
∑
v∈V1
avC
v
g (ψ1, . . . , ψs) stands for a linear combination of complete contractions in the
form:
contr
(
∇a1...at∇(p)
r1...rpψq ⊗∇
b1...by∇(q)
z1...zqψw ⊗R
2
⊗∆ψ1 ⊗ · · · ⊗ ∆̂ψq ⊗ · · · ⊗ ˆ∆ψw ⊗ · · · ⊗∆ψs
)
, (3.41)
with δ ≥ µ+ 2.
∑
v∈V2
avC
v
g (ψ1, . . . , ψs) is a linear combination of complete contractions in the
form:
contr
(
∇a1...at∇(p)
r1...rpψu ⊗∇
b1...by∇(q)
z1...zqRicij ⊗R⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψu ⊗ · · · ⊗∆ψs
)
, (3.42)
with δ ≥ µ+ 2.
∑
v∈V3
avC
v
g (ψ1, . . . , ψs) is a linear combination of complete contractions in the
form:
contr
(
∇a1...at∇(p)
r1...rpRicij ⊗∇b1...by∇(q)
z1...zqRici′j′ ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
, (3.43)
with δ ≥ µ+ 2. Finally, Cspec
g (ψ1, . . . , ψs) stands for a complete contraction in the form:
contr
(
∆(n
2
−σ−2)∇ijR⊗∇ijR⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
where δ ≥ µ+ 1.
(3.40) follows by the definition of
∑
f∈F · · · , apart for the claims regarding the numbers of
internal contractions in the different factors. We check these claims by virtue of our assumptions
on Isg . Firstly, polarizing the function ψ, we write out Isg as a linear combination of contractions
in the form:
contr
(
∇a1...at∇(m)Wijkl ⊗ · · · ⊗ ∇b1...bu∇(mσ−s)Wi′j′k′l′
⊗∇v1...vx∇(p1+2)ψ ⊗ · · · ⊗ ∇y1...yw∇(ps+2)ψ
)
. (3.44)
Now, using the decomposition of the Weyl tensor we decompose the above complete contractions.
As noted in the introduction, we observe that each complete contraction with a factor R must
have δ ≥ µ + 2, This shows our claim for (3.41) and (3.42). Moreover, we recall that all
complete contractions C l(g), l ∈ Θσ−2 with s factors P aa are in the form Cα, as in (3.39).
Hence, by applying the decomposition of the Weyl tensor to the complete contraction Cα, we
see that the contribution of Cα to the linear combination
∑
f∈F 2
σ−2
afC
f is precisely the complete
contraction Cspec (times a constant). For each other complete contraction C l, l ∈ Θσ−2\Θ∗σ−2, we
have that C lg(ψ1, . . . , ψs), written in the form (3.44) does not have s factors ∆ψh (by definition).
Hence, each complete contraction Cf , f ∈ F 2
σ−2, in the decomposition of each C lg(ψ1, . . . , ψs),
l ∈ Θσ−2 \Θ∗σ−2 must have δ ≥ µ+ 2. This shows (3.43).
In view of (3.40), and by applying the same “by hand” technique as for equation (3.4), we see
that we can find a vector field
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so that modulo complete contractions
of length ≥ σ + 1:∑
f∈F 2
σ−2
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) = (const)!C
!
g(ψ1, . . . , ψs)
+
σ−2∑
u=1
(const)!,uC
!,u
g (ψ1, . . . , ψs) +
σ−2∑
q=1
q−1∑
w=1
(const)!,(q,w)C
!,(q,w)
g (ψ1, . . . , ψs)
30 S. Alexakis
+
∑
p∈P 2,δ≥µ+2
σ−2
apC
p
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs), (3.45)
where
∑
p∈P 2,δ≥µ+2
σ−2
apC
p
g (ψ1, . . . , ψs) stands for a linear combination of complete contractions in
the form (1.4) with length σ, s = σ − 2, |∆| = σ − 2 and q = 1 and also δ ≥ µ + 2. The “by
hand” construction of the vector field for (3.45) is the same as the proof of equation (3.4), with
the slight caveat that we now also have the complete contraction Cspec. In particular, since we
only know that δ ≥ µ + 1 for Cspec, we have to check that we get no correction terms of the
form (1.4) with q = 1. But this follows by the form of Cspec, i.e. that we have two factors ∇(t)R,
so we never have to introduce correction terms by virtue of the second Bianchi identity.
A study of the sublinear combination
∑
f∈F 1
σ−2
afC
f
g (ψ1, . . . , ψs). We claim that
this sublinear combination will be of the form:∑
f∈F 1
σ−2
afC
f
g (ψ1, . . . , ψs) =
∑
d∈D
adC
d
g (ψ1, . . . , ψs)
+ (const)βC
β
g (ψ1, . . . , ψs) + (const)γC
γ
g (ψ1, . . . , ψs), (3.46)
where
∑
d∈D adC
d
g (ψ1, . . . , ψs) stands for a generic linear combination of complete contractions
in the form (1.4) with length σ, s = σ − 2, |∆| = σ − 2 and q = 1 and also δ ≥ µ + 2 (just
like the linear combination
∑
p∈P 2,δ≥µ+2
σ−2
apC
p
g (ψ1, . . . , ψs) in (3.45)), while Cβg (ψ1, . . . , ψs) is the
complete contraction:
contr
(
∆(n
2
−σ−2)∇ilRijkl ⊗∇jkR⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
, (3.47)
with δ = n
2 ≥ µ+ 1, and Cγg (ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−σ−2)∇jkR⊗∇ilRijkl ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
, (3.48)
with δ = n
2 ≥ µ+ 1.
Observe that the sublinear combination
(const)βC
β
g (ψ1, . . . , ψs) + (const)γC
γ
g (ψ1, . . . , ψs)
above is exclusively the contribution of the complete contraction Cα in Is to the linear combi-
nation
∑
f∈F 1
σ−2
afC
f
g (ψ1, . . . , ψs). (The fact that such contractions in the forms (3.47), (3.48)
do arise in the decomposition of Cα can be directly checked.)
Now, we check that the contribution of all the other complete contractions in Is (i.e. other
than Cα) to the sublinear combination
∑
f∈F 1
σ−2
afC
f
g (ψ1, . . . , ψs) is indeed
∑
d∈D adC
d
g (ψ1,
. . . , ψs), as described in (3.46).
But this is straightforward to observe. We only need to check that each C lg(ψ1, . . . , ψs) in
the form (3.44), with δ ≥ µ and with less than s factors ∆ψs can be written out as:
C lg(ψ1, . . . , ψs) =
∑
d∈D
adC
d
g (ψ1, . . . , ψs) +
2∑
q=0
∑
|∆|≤σ−3
∑
f∈F q|∆|
afC
f
g (ψ1, . . . , ψs), (3.49)
where
∑
d∈D adC
d
g (ψ1, . . . , ψs) is as above. (3.49) just follows by the decomposition of the Weyl
tensor.
Now, we seek to “get rid” of the complete contractions Cβ, Cγ . We construct vector fields
Cβ,i, Cγ,i, where Cβ,i arises from Cβ by erasing the index j in the second factor and making
the index j that it contracted against in the first factor into a free index i. Cγ,i arises from Cγ
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 31
by erasing the index j in the first factor and making the index j that it contracted against in
the second factor into a free index i.
We then observe that modulo complete contractions of length ≥ σ + 1:
Cβg (ψ1, . . . , ψs)− diviC
β,i
g (ψ1, . . . , ψs) =
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
Cγg (ψ1, . . . , ψs)− diviC
γ,i
g (ψ1, . . . , ψs) =
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs).
Next, we claim that we can subtract a divergence
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) from the linear
combination
∑
d∈D adC
d
g (ψ1, . . . , ψs) so that:∑
d∈D
adC
d
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) = (const)? · C?
g(ψ1, . . . , ψs)
+
s∑
u=1
(const)??,u · C??,u
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
where C?
g(ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−σ−1)∇ilRijkl ⊗ Ricjk ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
and it has δ ≥ µ+ 2, while C??,u
g (ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−σ−1)∇ilRijkl ⊗∇jkψu ⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψu ⊗ · · · ⊗∆ψs
)
,
and it has δ ≥ µ+2. This follows by imitating the proof of the previous case. Then, we explicitly
construct a vector field
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so that:
(const)? · C?
g(ψ1, . . . , ψs) +
s∑
u=1
(const)??,u · C??,u
g (ψ1, . . . , ψs)
− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) = (const)] · C]g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
where C]g(ψ1, . . . , ψs) stands for the complete contraction:
contr
(
∆(n
2
−σ−2)∇ilRijkl ⊗∇i
′l′Ri′
jk
l′ ⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
which has δ ≥ µ+ 1.
Now, recall that by our lemma’s assumption L∗µ = ∅, and
∑
j∈J∗ ajC
j
g(ψ1, . . . , ψs) =
(const)′′C]g(ψ1, . . . , ψs). Therefore, by our study of the three sublinear combinations∑
f∈F 1
σ−2
· · · ,
∑
f∈F 2
σ−2
· · · ,
∑
f∈F 2
σ−1
· · · we have shown that we can subtract a divergence from Is
so that:
Isg(ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
∑
l∈Lµ
alC
l
g(ψ1, . . . , ψs)
+
∑
j∈J\J∗
ajC
j
g(ψ1, . . . , ψs) +
2∑
q1=0
σ−3∑
v=0
∑
f∈F q1v
afC
f
g (ψ1, . . . , ψs) + (const)! · C !
g(ψ1, . . . , ψs)
+
s∑
y=1
(const)y,! · Cy,∗g (ψ1, . . . , ψs) +
s∑
w=1
w−1∑
q=1
(const)w,q · C !,w,q
g (ψ1, . . . , ψs)
32 S. Alexakis
+ (const)]C
]
g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs).
Here each C l, Cj has |∆| ≤ σ − 3. We then apply the exact same proof as in the third step in
the previous subsection to show that (const)! = 0, (const)y,! = 0, (const)w,q = 0, (const)] = 0.
We have thus shown our claim in the case s = σ − 2, µ < n
2 . �
The proof of Lemmas 1.2, 1.3, 1.4 when s = σ − 2, σ < n
2
−1 and µ = n
2
. In this ca-
se, we have that P (g)|Θσ−2 can be written out as follows:
P (g)|Θσ−2 =
∑
l∈L
alC
l(g),
where each C l(g) is a complete contraction in the form:
contr
(
∆(α1)∇ilWijkl ⊗∆(α2)∇i′l′W i′jkl′ ⊗∆(α3)P aa ⊗ · · · ⊗∆(ασ)P aa
)
. (3.50)
This is true by virtue of the fact that each C l(g), l ∈ Θσ−2 must have δ = n
2 . We assume
with no loss of generality that the last σ − 2 factors are arranged so that α3 ≥ α4 ≥ · · · ≥ ασ.
We denote by L0 ⊂ L the index set of the complete contractions C l(g) with α3 = · · · = ασ = 0.
We claim that:∑
l∈L0
alC
l(g) = 0. (3.51)
We observe that if we can show this, we will then have shown our claim. This is because
every complete contraction C l(g), l ∈ L \ L0 will have a factor ∆(α)P aa with α > 0, hence each
complete contraction C l, Cj , Cf in (1.5) will have a factor ∇(p)ψh with p ≥ 3 and two curvature
factors that are not R (scalar curvature). Hence, if we can show (3.51), we will have shown the
remaining cases of both our Lemmas when µ = n
2 and σ < n
2 − 1.
We will, in fact show a more general statement: We denote by Lγ ⊂ L the subsets of complete
contractions in the form with α3 = γ, α4 = · · · = ασ = 0. We will then show that for each
γ ≥ 0:∑
l∈Lγ
alC
l(g) = 0. (3.52)
We show the above by an induction. Let us assume that (3.52) is known for every γ > γ1.
We will then show (3.52) for γ = γ1. By our inductive assumption, we may cross out from P (g)
the sublinear combination
∑
γ>γ1
∑
l∈Lγ alC
l(g). Now, we consider Isg(ψ1, . . . , ψs). We pick out
the sublinear combination of Is,γ1
g (ψ1, . . . , ψs) of complete contractions in the form:
contr
(
∆(α1)∇ilWijkl ⊗∆(α2)∇i′l′W i′jkl′ ⊗∆(γ1+1)ψ1 ⊗∆ψ2 ⊗ · · · ⊗∆ψσ−2
)
(3.53)
(we make the convention that α1 ≥ α2).
Clearly, if we can show that modulo complete contractions of length ≥ σ + 1:
Is,γ1
g (ψ1, . . . , ψs) = 0,
we will then have shown our inductive claim.
By the transformation law of the Schouten tensor under conformal re-scalings (see [3]), and
by virtue of our inductive hypothesis (3.50) for γ > γ1, we deduce that we can write out:
Isg(ψ1, . . . , ψs) = Is,γ1
g (ψ1, . . . , ψs) +
∑
u∈U1
auC
u
g (ψ1, . . . , ψs) +
∑
u∈U2
auC
u
g (ψ1, . . . , ψs), (3.54)
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 33
where each Cu is in the form (3.50), and for u ∈ U1 αh > 0 for at least one h > 0, while for each
u ∈ U2 we have that α3 < γ1.
We now use the silly divergence formula for Isg (recall this formula from [1]), by integrating by
parts with respect to the derivatives on the factor∇(α)ψ1. We denote the silly divergence formula
for Is by silly[Isg ]. We focus on the sublinear combination silly∗[I
s
g ] of complete contractions in
silly[Isg ] in the form:
contr
(
∇t1...tγ1+1∆(α1)∇ilWijkl ⊗∇t1...tγ1+1∆(α2)∇i′l′W i′jkl′
⊗ ψ1 ⊗∆ψ2 ⊗ · · · ⊗∆ψσ−2
)
. (3.55)
Now, let us write out Is,γ1
g (ψ1, . . . , ψs) in the form:
Is,γ1
g (ψ1, . . . , ψs) =
∑
(p,q)
a(p,q)C
(p,q)
g (ψ1, . . . , ψs),
where C(p,q) stands for the complete contraction in the form (3.53) with α1 = p, α2 = q. We
then denote by C
(p,q),]
g the complete contraction that arises from C(p,q) by replacing the factor
∆(γ1+1)ψ1 by a factor ψ1 (with no derivatives) and hitting the factor ∆(p)∇ilWijkl by derivatives
∇t1...tγ1+1 and the factor ∆(q)∇i′l′W i′jkl′ by derivatives ∇t1...tγ1+1 (we are using the Einstein
summation convention). We then make three claims. Firstly:
silly∗[I
s
g ] = 2γ1
∑
(p,q)
a(p,q)C
(p,q),]
g (ψ1, . . . , ψs).
Now, for each C(p,q),] above, we denote by C
(p,q),],ι
g (ψ1, . . . , ψs) the complete contraction that
arises from it by replacing the two factors ∇t1...tγ1+1∆(α1)∇ilWijkl, ∇t1...tγ1+1∆(α2)∇i′l′W i′jkl′
by factors ∇t1...tγ1+1∆(α1)∇ilRijkl, ∇t1...tγ1+1∆(α2)∇i′l′Ri
′jkl′ . Our second claim is that modulo
complete contractions of length ≥ σ + 1:∑
(p,q)
a(p,q)C
(p,q),],ι
g (ψ1, . . . , ψs) = 0.
Our third claim is that from the above we can deduce that each a(p,q) = 0. If we can show
the above three claims, then clearly (3.52) with γ = γ1 will follow.
We begin with our first claim. For each complete contraction Cg(ψ1, . . . , ψs) in (3.54), we
denote by sil[Cg(ψ1, . . . , ψs)] ·ψ1 the sublinear combination of complete contractions in the right
hand side of silly[Isg ]. This notation extends to linear combinations. The silly divergence formula
just tells us that:
sil[Is,γ1
g (ψ1, . . . , ψs)] +
∑
u∈U1
ausil[Cug (ψ1, . . . , ψs)] +
∑
u∈U2
ausil[Cug (ψ1, . . . , ψs)] = 0.
If we can show that for u ∈ U1
⋃
U2, sil[Cug ] contains no complete contractions in the
form (3.55), our first claim will then follow.
This claim is trivial for the complete contractions Cug , u ∈ U1. We notice that if Cug has
a factor ∆(α)ψh, α > 1, h > 1, then each complete contraction in sil[Cug ] will have a factor
∇(p)ψh, p ≥ 2α > 2. On the other hand, for each u ∈ U2, we observe that each complete
contraction in sil[Cug ] will have less than γ1 + 3 indices in the first factor ∇(m)Wijkl contracting
against indices in the second factor ∇(m′)Wijkl. So we indeed see that sil[Cug ], u ∈ U2 has no
complete contraction of the form (3.55). Therefore we have shown our first claim.
34 S. Alexakis
Now, we check the second claim. We write out silly[Isg(ψ1, . . . , ψs)] as a linear combination
of complete contractions in the form (1.4). We define sillyspec[I
s
g(ψ1, . . . , ψs)] to stand for the
sublinear combination of complete contractions with s− 1 factors ∆ψ2, . . . ,∆ψs, one factor ψ1
and two factors ∇(m)Rijkl, ∇(m′)Ri′j′k′l′ with γ1 + 3 particular contractions between them. (We
write ∇(m)Rijkl, ∇(m′)Ri′j′k′l′ but there is no restriction on the particular contractions of the
indices i, j , k, l, i′ , j′ , k′ , l′ , i.e. we can have i, k contracting between themselves etc.) Since the silly
divergence formula must hold formally, we deduce that:
sillyspec[I
s
g(ψ1, . . . , ψs)] = 0, (3.56)
modulo complete contractions of length ≥ σ + 1.
Moreover, clearly any complete contraction Cg(ψ1, . . . , ψs) in the form (3.50) with either
a factor ∇(p)ψh, p ≥ 3 or with less than γ1 + 3 pairs of particular contractions between
the first two factors ∇(m)Wijkl, ∇(m′)Wijkl cannot contribute to the sublinear combination
sillyspec[I
s
g(ψ1, . . . , ψs)]. Hence, we derive that sil[Cug ], u ∈ U1
⋃
U2 do not contribute to the sub-
linear combination sillyspec[I
s
g ], whereas by virtue of the decomposition of the factor ∇ilWijkl,
we have that:
sillyspec[I
s
g(ψ1, . . . , ψs)] =
∑
(p,q)
2γ1+1n− 3
n− 2
a(p,q)C
(p,q),],ι
g (ψ1, . . . , ψs) +
∑
t∈T
atC
t
g(ψ1, . . . , ψs),
where each is in the form (1.4) with either two factors of the form ∇(p)R, ∇(p′)R,33 or one factor
of the form ∇(p)R and one factor of the form ∇(m)Rijkl where none of the indices i, j , k, l are
contracting between themselves. In fact, in order to distinguish these two cases we break the
index set T into subsets T1, T2 accordingly.
Now, since (3.56) holds formally we deduce that modulo complete contractions of length
≥ σ + 1:∑
t∈T1
atC
t
g(ψ1, . . . , ψs) = 0.
Then, again since (3.56) holds formally, we deduce that modulo complete contractions of length
≥ σ + 1:∑
t∈T2
atC
t
g(ψ1, . . . , ψs) = 0.
Therefore, we have shown that modulo complete contractions of length ≥ σ + 1:∑
(p,q)
a(p,q)C
(p,q),],ι
g (ψ1, . . . , ψs) = 0.
Clearly, since the above holds formally, we can deduce that for each different pair (p, q) we
must have:
a(p,q)C
(p,q),],ι
g (ψ1, . . . , ψs) = 0,
and since each complete contraction C
(p,q),],ι
g (ψ1, . . . , ψs) is not identically the zero contraction,
we deduce that each a(p,q) must be zero. This was our third claim. We have thus shown our
inductive statement, and we have proven our claim in the case s = σ− 2, σ < n
2 − 1, µ = n
2 . �
The proof of Lemmas 1.2, 1.3, 1.4 when s = σ − 2 and σ = n
2
− 1. We observe that
any complete contraction C l(g) in P (g)|σ, when σ = n
2 − 1 must be in the form:
contr
(
∇(m)Wijkl ⊗∇(m′)Wi′j′k′l′ ⊗∇(p1)Pab ⊗ · · · ⊗ ∇
(pn
2−3)
Pa′b′
)
must have exactly two derivatives (i.e. m+m′+ p1 + · · ·+ pn
2
−3 = 2) and moreover δ ≤ n
2 − 1.34
33Recall that R here is the scalar curvature.
34It follows easily that in this case µ ≤ n
2
.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 35
We distinguish two cases. Either µ = n
2 − 1 or µ < n
2 − 1. We start with the first case. Then
we see that:
P (g)|Θσ−2 = (const) · C(g),
where C(g) is the complete contraction:
contr
(
∇lWijkl ⊗∇l′W ijkl′ ⊗ (P aa )σ−2
)
. (3.57)
We will show that (const) = 0. This follows easily by the silly divergence formula. We
consider Isg(ψ1, . . . , ψs). It follows that:
Isg(ψ1, . . . , ψs) = s! · Cg(ψ1, . . . , ψs),
where Cg(ψ1, . . . , ψs) is the contraction:
contr
(
∇lWijkl ⊗∇l′W ijkl′ ⊗∆ψ1 ⊗ · · · ⊗∆ψσ−2
)
.
We consider silly[Isg ], with integration by parts with respect to ∇(p)ψ1, and we focus on the
sublinear combination:
silly+[Isg(ψ1, . . . , ψs) = (const)′ ·contr
(
∇laWijkl ⊗∇l′aW ijkl′ ⊗ ψ1 ⊗∆ψ2 ⊗ · · · ⊗∆ψσ−2
)
.
It follows by the same arguments as before that silly+[Isg ] = 0, modulo complete contractions
of length ≥ σ + 1 and that (const)′ = 2 · s!(const), and hence we deduce our claim in the case
µ = n
2 − 1.
Now, the case µ < n
2 − 1. We notice that in this case, the sublinear combination of complete
contractions C l(g), l ∈ Θσ−2 that have σ − 2 factors P aa is precisely of the form (const) · C(g),
where C(g) is in the form (3.57) (since P (g)|Θσ−1 is assumed “good”). We can then prove our
claim in this case by exactly applying the method of the case σ = n
2 − 1 when s < σ − 2. �
3.3 The proof of Lemmas 1.2, 1.3, 1.4 when s = σ − 1,
and Lemma 1.5 when s = σ − 1
In this case, we follow the same pattern as in the two previous ones. We begin by a trivial
observation regarding the maximum value that µ can have35. By virtue of the formula Iσ−1
g (φ) :=
dσ−1
dtσ−1 |t=0e
ntφP (e2tφg),36 and since we observe that any complete contraction in the form (1.3)
with length σ and σ−1 factors P aa must be zero, modulo complete contractions of length ≥ σ+1,
we observe that necessarily µ ≤ n
2 − 1 in this case.
We again distinguish the two cases µ = n
2 − 1 and µ < n
2 − 1. We start with the second case.
Proof of Lemma 1.5 when s = σ − 1, µ < n
2
− 1. We focus on the sublinear combination
of complete contractions C l(g) in P (g)|Θσ−1 that have σ − 2 factors P aa . We denote their index
set by Θ+
σ−1 ⊂ Θσ−1. By the usual “manual” construction of explicit divergences it follows
that we can subtract a divergence divi
∑
h∈H ahC
h,i(g) from P (g)|Θ+
σ−1
so that modulo complete
contractions of length ≥ σ + 1:
P (g)|Θ+
σ−1
− divi
∑
h∈H
ahC
h,i(g) = (const)V · CV (g) +
∑
t∈T
atC
t(g) +
∑
y∈Y
ayC
y(g),
35Recall that µ stands for the minimum value of δ = δW + δP among the complete contractions Cl(g), l ∈ Θs.
36Thus in particular the terms of length σ − 1 in Iσ−1
g (φ) arise from the sublinear combination P (g)Θσ−1 by
just replacing each factor ∇(a)
r1...raPij by −∇(a+2)
r1...raij
φ.
36 S. Alexakis
where CV (g) is the complete contraction:
contr
(
∆
n
2
−(σ−2)−3∇ilWijkl ⊗ P jk ⊗ (P aa )σ−2
)
,
and where each Cy(g) is in the form (1.3) with at least two factors ∇(m)Wijkl. Also, each Ct(g)
is in the form (1.3) with δ ≥ µ and also with less than σ − 2 factors P aa .
Now, we observe that CV (g) has δ = n
2 − 1 ≥ µ+ 1. Therefore, if we define CV,i(g) to stand
for the vector field that arises from CV (g) by erasing i and making i in the first factor into
a free index, we will then have:
CV (g)− diviC
V,i(g) =
∑
y∈Y
ayC
y(g) +
∑
d∈D
adC
d(g).
Here each Cy(g) is as above, while each Cd(g) has δ ≥ µ and also has strictly less than σ − 2
factors P aa . Thus, by subtracting the divergence divi
[∑
h∈H ahC
h,i(g) + (const)V · CV,i(g)
]
from P (g), we may assume with no loss of generality that P (g)|Θσ−1 contains only complete
contractions with at most σ − 3 factors P aa . We will be using this fact below. �
Proof of Lemmas 1.2, 1.3, 1.4 when s = σ−1, µ < n
2
−1. We focus on Isg(ψ1, . . ., ψs);
37
for the global conformal invariants P (g) above, and we decompose the Weyl tensors (i.e. we
write Isg as a linear combination in the form (1.5)). In the notation of the introduction, we have
that F ∗ = F 1
σ−1
⋃
F 1
σ−2.
We first focus our attention on the sublinear combination F 1
σ−1. As in the previous cases, it
follows that we can write out:∑
f∈F 1
σ−1
afC
f
g (ψ1, . . . , ψs) = (const)+ · C+
g (ψ1, . . . , ψs) +
s∑
u=1
auC
u
g (ψ1, . . . , ψs),
where C+
g (ψ1, . . . , ψs) stands for the complete contraction:
contr
(
∆(n
2
−σ)R⊗∆ψ1 ⊗ · · · ⊗∆ψs
)
,
while Cug (ψ1, . . . , ψs) stands for the complete contraction:
contr
(
∆(n
2
−σ+1)ψu ⊗∆ψ1 ⊗ . . . ˆ∆ψu · · · ⊗∆ψs ⊗R
)
.
We observe that each of the above complete contraction has δ = n
2 + 1 ≥ µ + 2. As before,
we can explicitly construct a divergence divi
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) such that:
∑
f∈F 1
σ−1
afC
f
g (ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
s∑
u=1
(const)∗,uC
∗,u
g (ψ1, . . . , ψs)
+
s∑
q=1
q−1∑
w=1
(const)∗,(q,w)C
∗,(q,w)
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
where C∗,ug (ψ1, . . . , ψs) stands for the complete contraction:
contr
(
∆(n
2
−σ−1)R⊗∆(2)ψu ⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψu ⊗ · · · ⊗∆ψs
)
,
37Recall that Isg(φ) := ds
dts
|t=0(entφP 2tφg) and that Isg(ψ1, . . . , ψs) arises from Isg(φ) by just polarizing the
function φ.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 37
while C
∗,(q,w)
g (ψ1, . . . , ψs) stands for the complete contraction
contr
(
∆(n
2
−σ)ψq ⊗∆(2)ψw ⊗∆ψ1 ⊗ · · · ⊗ ∆̂ψq ⊗ · · · ⊗ ˆ∆ψw ⊗ · · · ⊗∆ψs
)
,
and
∑
z∈Z′ azC
z
g (ψ1, . . . , ψs) stands for a generic linear combination of complete contractions
with |∆| ≤ σ − 3 and δ ≥ µ+ 1.
We next focus our attention on the sublinear combination
∑
f∈F 1
σ−2
afC
f
g (ψ1, . . . , ψs). Since
P (g)|Θσ−1 has no complete contractions with σ−2 factors P aa , it follows that any Cfg (ψ1, . . . , ψs),
f ∈ F 1
σ−2 must have δ ≥ µ+2. Therefore, as in the previous subsection, we can construct a vector
field
∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so that, modulo complete contractions of length ≥ σ + 1:
∑
f∈F 1
σ−2
afC
f
g (ψ1, . . . , ψs)−
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs) =
s∑
u=1
(const)′∗,uC
∗,u
g (ψ1, . . . , ψs)
+
s∑
q=1
q−1∑
w=1
(const)′∗,(q,w)C
∗,(q,w)
g (ψ1, . . . , ψs) +
∑
b∈B
abC
b
g(ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs).
Here the complete contractions C∗,u, C∗,(q,w) are the same as above, while
∑
b∈B abC
b
g(ψ1, . . . , ψs)
stands for a linear combination of complete contractions in the form (1.4) with length σ, δ ≥ µ+2,
q = 0 and |∆| = σ − 2.
Finally, we focus on the sublinear combinations in Isg(ψ1, . . . , ψs) which have q = 0 (i.e. we
consider the complete contractions that are indexed in the sets Lµ, J in (1.5)38). Now, since we
have that each C l(g), l ∈ Θσ−1 has less than σ − 2 factors P aa , it follows that all the complete
contractions in Lµ, J in (1.5) must have |∆| ≤ σ − 3.
Then (by the usual construction), we can construct a linear combination of vector fields,∑
h∈H ahC
h,i
g (ψ1, . . . , ψs) so that, modulo complete contractions of length ≥ σ + 1:∑
b∈B
abC
b
g(ψ1, . . . , ψs)− divi
∑
h∈H
ahC
h,i
g (ψ1, . . . , ψs)
=
s∑
u=1
(const)L,uC
L,u
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs),
where CL,ug (ψ1, . . . , ψs) is the complete contraction:
contr
(
∆(n
2
−(σ−2)−3)∇ilRijkl ⊗∇jkψu ⊗∆ψ1 ⊗ · · · ⊗ ˆ∆ψu ⊗ · · · ⊗∆ψs
)
,
which has δ ≥ µ+ 2, and where
∑
z∈Z′ azC
z
g (ψ1, . . . , ψs) is as above.
Moreover, we define CL,u,ig (ψ1, . . . , ψs) to stand for the vector field that arises from CL,ug (ψ1,
. . . , ψs) by erasing the index i and making i into a free index. We then observe that, modulo
complete contractions of length ≥ σ + 1:
CL,ug (ψ1, . . . , ψs)− diviC
L,u,i
g (ψ1, . . . , ψs) =
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs).
In conclusion, we have shown that we can subtract a linear combination of divergences from
Isg(ψ1, . . . , ψs) so that, modulo complete contractions of length ≥ σ + 1:
Isg(ψ1, . . . , ψs)− diviC
h,i
g (ψ1, . . . , ψs) =
∑
l∈Lµ
alC
l,ι
g (ψ1, . . . , ψs) +
∑
j∈J
ajC
j
g(ψ1, . . . , ψs)
38These sets index the complete contractions in Isg with no factors ∇(p)Ric.
38 S. Alexakis
+
∑
f∈F 1\(F 1
σ−1
⋃
F 1
σ−2)
afC
f
g (ψ1, . . . , ψs) +
s∑
u=1
(const)∗,uC
∗,u
g (ψ1, . . . , ψs)
+
s∑
q=1
q−1∑
w=1
(const)∗,(q,w)C
∗,(q,w)
g (ψ1, . . . , ψs) +
∑
z∈Z′
azC
z
g (ψ1, . . . , ψs).
(The terms in the second line are not generic linear combinations – they stand for the original
linear combinations in Isg(ψ1, . . . , ψs) as in (1.5).)
Then, using the super divergence formula as in the previous subsection, we deduce that
(const)∗,u = 0 for every u = 1, . . . , s and (const)∗,(q,w) = 0 for every q = 1, . . . , s, w = 1, . . . , q−1.
Therefore, we have proven Lemmas 1.1, 1.3, 1.4 when s = σ − 1 and µ < n
2 − 1. �
Now, we consider the subcase where µ = n
2 −1. We see that in this case (modulo introducing
correction terms with length σ and two factors ∇(m)Wijkl), we can write out P (g)|Θσ−1 :
P (g)|Θσ−1 =
∑
γ∈Γ
aγC
γ
g (ψ1, . . . , ψs) +
∑
ε∈E
aεC
ε
g(ψ1, . . . , ψs),
where each Cγ(g) is in the form:
contr
(
∆(α)∇ilWijkl ⊗∆(β)P jk ⊗∆(ρ1)P aa ⊗ · · · ⊗∆(ρσ−2)P aa
)
(3.58)
(we are making the convention that ρ1 ≥ · · · ≥ ρσ−2), while each Cε(g) is in the form:
contr
(
∆(α)∇ilWijkl ⊗∆(β1)∇tP jt ⊗∆(β2)∇wP kw ⊗∆(ρ1)P aa ⊗ · · · ⊗∆(ρσ−3)P aa
)
(3.59)
(we are making the convention that ρ1 ≥ · · · ≥ ρσ−3).
Now, a small observation. For each Cγ(g) with β > 0, we can construct “by hand” a vector
field Cγ,i(g) so that:
Cγ(g)− diviC
γ,i(g) =
∑
y∈Y
ayC
y(g) +
∑
t∈T
atC
t(g), (3.60)
where each Cy(g) is a complete contraction in the form (3.59) and
∑
t∈T atC
t(g) is a linear
combination of complete contractions in the form (1.3) with at least two factors ∇(m)Wijkl. The
above holds modulo complete contractions of length ≥ σ + 1.
In view of (3.60), we can assume that each Cγ(g) in P (g)|Θσ−1 has β = 0.
We then define Γκ to stand for the index set of the complete contractions in the form (3.58)
with β = 0, ρ1 = κ ≥ 0 and ρ2, . . . , ρσ−2 = 0. We observe that, by definition:∑
γ∈Γκ
aγC
γ(g) = (const)κ · Cκ(g),
where Cκ(g) is in the form (3.58) with β = 0, ρ1 = κ, ρ2 = · · · = ρs−2 = 0 and with α =
n
2 − (σ − 2)− 3.
We will show that under the above assumptions, for each κ ≥ 0:∑
γ∈Γκ
aγC
γ
g (ψ1, . . . , ψs) = 0, (3.61)
modulo complete contractions of length ≥ σ + 1.
If we can show the above, we then see that we will have shown Lemmas 1.1, 1.3, 1.4 in
this setting. This is true since if we consider any P (g) with P (g)|Θσ−1 that satisfies (3.61)
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 39
and we then write out Isg(ψ1, . . . , ψs) by decomposing the Weyl tensors, then each complete
contraction C l,ι, Cf , Cj in Is must have at least two factors ∇(t)ψh with t ≥ 3 and at least one
factor either in the form ∇(m)Rijkl, m ≥ 2 or in the form ∇(p)Ric, p ≥ 2.
We show (3.61) by an induction. We assume that (3.61) holds for each κ > κ1 and we will
show it for κ = κ1. We consider the silly divergence formula for Isg(ψ1, . . . , ψs) – we denote it by
silly[Isg(ψ1, . . . , ψs)] – and we focus on the sublinear combination silly∗[I
s
g ] that consists of the
complete contractions in the form:
contr
(
∇t1...tκ1
∆(n
2
−(σ−2)−3−κ1)∇ilRijkl ⊗∇t1...tκ1jkψ1 ⊗ ψ2 ⊗∆ψ3 ⊗ · · · ⊗∆ψs−2
)
. (3.62)
We then make two claims. Firstly, that modulo complete contractions of length ≥ σ + 1:
silly∗[I
s
g(ψ1, . . . , ψs)] = 0,
and secondly that:
silly∗[I
s
g(ψ1, . . . , ψs)] = 2κ1
n− 3
n− 2
(const)κ1 · Cκ1
g (ψ1, . . . , ψs),
where Cκ1
g is in the form (3.62). Since Cκ1 is clearly not identically zero, we will then deduce
that (const)κ1 = 0, and we will have shown our inductive claim.
We begin with the first claim. We initially denote by silly+[Isg ] the sublinear combina-
tion in silly[Isg ] that consists of complete contractions with length σ and a factor ψ2, factors
∆ψ3, . . . ,∆ψs−2, a factor ∇(κ1+2)ψ1 with no internal contractions and a factor ∇(m)Rijkl (where
now we are allowing any two indices in this last factor to be contracting against each other). In
other words, we are looking at the sublinear combination of complete contractions in the form:
contr
(
∇(m)Rijkl ⊗∇(κ1+2)ψ1 ⊗ ψ2 ⊗∆ψ3 ⊗ · · · ⊗∆ψs
)
, (3.63)
where ∇(κ1+2)ψ1 has no internal contraction.
Since the silly divergence formula holds formally, we deduce that:
silly+[Isg ] = 0,
modulo complete contractions of length ≥ σ + 1.
We are now in a position to show our first claim above. By just applying the decomposition
of the Weyl tensor, we calculate:
silly+[Isg(ψ1, . . . , ψs)] = silly∗[I
s
g(ψ1, . . . , ψs)] + (const)′ · C ′g(ψ1, . . . , ψs), (3.64)
where C ′g(ψ1, . . . , ψs) is the complete contraction:
contr
(
∇t1...tκ1+2∆(power)R⊗∇t1...tκ1+2ψ1 ⊗ ψ2 ⊗∆ψ3 ⊗ · · · ⊗∆ψs
)
.
Now since silly+[Isg(ψ1, . . . , ψs)] = 0, we may derive that the second sublinear combination
in (3.64) vanishes separately39. Hence we derive our first claim, that silly∗[I
s
g(ψ1, . . . , ψs)] = 0.
We now proceed to the second claim. We would like to understand how the sublinear combi-
nation silly+[Isg ] is related to Isg(ψ1, . . . , ψs). Given that µ = n
2 − 1 in our case, while silly+[Isg ]
consists of complete contractions with factors ∆ψ3, . . . ,∆ψs−1, it follows that a complete con-
traction Cg(ψ1, . . . , ψs−1) in Isg (where Cg is in the form (3.44)) can only contribute to silly+[Isg ]
if it has factors ∆ψ3, . . . ,∆ψs−1. (Otherwise, it must have at least one factor ∆yψv, v ≥ 3 with
y ≥ 2). Moreover, since the complete contractions in silly+[Isg ] each have a factor ∇(κ1+2)ψ1 with
39Using the operation Subω, defined in the section on technical tools, in the Appendix of [3].
40 S. Alexakis
no internal contractions, it follows that a complete contraction Cg(ψ1, . . . , ψs−1) in Isg (where Cg
is in the form (3.44)) can only contribute to silly+[Isg ] if it has a factor ∇(y)ψ1 with no internal
contractions (in the form (3.44)). Moreover, since we are considering complete contractions in
Isg(ψ1, . . . , ψs) with δ = n
2 − 1, it follows that y = 2, and therefore the two indices a, b in the
factor∇(2)
ab ψ1 must be contracting against two internal indices in the one factor∇(m)Wijkl (other-
wise, if they contracted against at least one derivative index, we must have at least one pair of
antisymmetric indices i, j or k, l in ∇(m)Wijkl both involved in an internal contraction, but such
a complete contraction must clearly be zero modulo complete contractions of length ≥ σ + 1).
Therefore, we have that the complete contractions in Isg(ψ1, . . . , ψs) that contribute to
silly+[Isg(ψ1, . . . , ψs)] must be in the form:
contr
(
∆(α)∇ilWijkl ⊗∇jkψ1 ⊗∆(β)ψ2 ⊗∆ψ3 ⊗ · · · ⊗∆ψs
)
.
We denote the above complete contraction by Cα,βg (ψ1, . . . , ψs). We recall that by our inductive
assumption, each Cα,βg in Isg must have β ≤ κ1.
Clearly, we also observe that for each Cα,βg above, the contribution of Cα,βg to silly+[Isg ] can
only be in silly[Cα,βg ] if we integrate by parts all the 2β indices (a1 , a1), . . . , (aβ , aβ ) in the factor
∆(β)ψ2 and we make the derivatives ∇a1 , . . . ,∇aβ hit the factor ∇(m)Rijkl and the derivatives
∇a1 , . . . ,∇aβ hit the factor ∇(2)ψ1. We observe that otherwise, we will either have an internal
contraction in the factor ∇(p)ψ1, or a factor ∇(p)ψh, p, h ≥ 3, or a factor ∇(t)ψ2, t ≥ 1, or
a factor ∇(y)ψ1 with an internal contraction.
We denote the complete contraction that we thus obtain by Ĉα,βg . Moreover, we clearly
observe that unless β = κ1, Ĉα,βg does not belong to silly+[Isg ]. To summarize, we have seen
that silly+[Isg ] is the sublinear combination in
∑
γ∈Γκ1
aγC
γ
g (ψ1, . . . , ψs) that consists of complete
contractions in the form (3.63).
We will now demonstrate our second claim. We employ the decomposition:
∇ilWijkl =
n− 3
n− 2
∇ilRijkl + (const)1∇2
jkR+ (const)2∆Rgjk.
We deduce that:
silly∗[I
s
g(ψ1, . . . , ψs)] =
∑
γ∈Γκ1
aγ
n− 3
n− 2
Cγ,ιg (ψ1, . . . , ψs).
This shows our second claim.
Acknowledgements
This work has absorbed the best part of the author’s energy over many years. This research
was partially conducted during the period the author served as a Clay Research Fellow, an
MSRI postdoctoral fellow, a Clay Liftoff fellow and a Procter Fellow. The author is immensely
indebted to Charles Fefferman for devoting twelve long months to the meticulous proof-reading
of the present paper. He also wishes to express his gratitude to the Mathematics Department
of Princeton University for its support during his work on this project.
The Decomposition of Global Conformal Invariants: Some Technical Proofs. I 41
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1 Introduction
2 Proof of Lemma 1.1
3 The proof of Lemmas 1.2, 1.3, 1.4, 1.5
3.1 The proof of Lemmas 1.2, 1.3, 1.4 when s<-2
3.2 The proof of Lemmas 1.2, 1.3, 1.4 and Lemma 1.5 when s=-2
3.3 The proof of Lemmas 1.2, 1.3, 1.4 when s=-1, and Lemma 1.5 when s=-1
References
|