CR Invariant powers of the sub-Laplacian

31 Pages·0.29 MB·English

by  A. Rod Gover

CR INVARIANT POWERS OF THE SUB-LAPLACIAN
3
0 A. ROD GOVER AND C. ROBIN GRAHAM
0
2
Abstract. CR invariant differential operators on densities with
n
leading parta powerofthe sub-Laplacianare derived. One family
a
J of such operators is constructed from the “conformally invariant
powers of the Laplacian” via the Fefferman metric; the powers
9
whicharisefortheseoperatorsareboundedintermsofthedimen-

] sion. A second family is derived from a CR tractor calculus which
G
is developed here; this family includes operators for every posi-
D
tive powerof the sub-Laplacian. This resulttogetherwith workof
. C˘ap, Slov´ak and Sou˘cek imply in three dimensions the existence
h
t of a curved analogue of each such operator in flat space.
a
m
[

1 1. Introduction
v
2 Invariant differential operatorshave alonghistoryofimportanceand
9
this is particularly the case for operators of Laplace type. The confor-
0
1 mally invariant Laplacian is the basic example in conformal geometry.
0 AfamilyofhigherordergeneralizationsoftheconformalLaplacianwith
3
principal part a power of the Laplacian was constructed in [22] and has
0
/ been the subject of recent interest. In CR geometry, the CR invariant
h
t sub-Laplacian of Jerison-Lee ([27]) plays a role analogousto that of the
a
m conformal Laplacian. In this paper we construct and study generaliza-
tions of the Jerison-Lee sub-Laplacian which are CR analogues of the
:
v
“conformally invariant powers of the Laplacian”.
i
X One can deduce the existence of some such operators in the CR
r case from the conformal operators via the Fefferman metric. Fefferman
a
[12] showed that to a CR manifold M of dimension 2n + 1 one can
associate a conformal structure on a circle bundle C of dimension N =
2n + 2. On conformal manifolds of dimension N, the construction
in [22] produces for each w, such that N/2 + w = k ∈ N and k ≤
N/2 if N is even, a conformally invariant natural differential operator
P : E(w) → E(w−2k), with principal part ∆k, where E(w) denotes
k
This research was partially supported by the Australian Research Council and
NSF grant DMS 0204480. This support and the hospitality of MSRI and the
Universities of Auckland and Washington are gratefully acknowledged.
1991 Mathematics Subject Classification. Primary 32V05; Secondary
53C07,53B15.
2 A.R.GOVER AND C. R. GRAHAM
the space of conformal densities of weight w. As we explain in the
next section, on a CR manifold one can consider CR densities E(w,w′)
parametrized by (bi-)weights (w,w′) ∈ C × C. It is necessary that
w − w′ ∈ Z in order that E(w,w′) be well-defined and this is always
implicitly assumed. The space E(w,w′) of CR densities on M can
be naturally regarded as a subspace of the space of conformal densities
E(w+w′)onC. TheoperatorsP fortheFeffermanmetriccanbeshown
k
to preserve the subspaces of CR densities and it is straightforward to
calculate the principal part of the induced operators. One thereby
obtains:
Theorem 1.1. Let n+1+w +w′ = k ∈ N with k ≤ n+1. There is
a CR invariant natural differential operator
Pw,w′ : E(w,w′) → E(w−k,w′ −k)
whose principal part agrees with that of ∆k.
b
In the case that M is a flat CR manifold, the hypothesis k ≤ n + 1
is unnecessary in this construction and in the flat case the operators
Pw,w′ exist for all (w,w′) such that n+1+w +w′ ∈ N.
Intheconformalcase,itisconjecturedthatifN isevenandk > N/2,
there is no natural operator with principal part ∆k mapping E(−N/2+
k) → E(−N/2−k). This has been established for N = 4, k = 3 ([21])
and N = 6, k = 4 ([35]). Our main result is the existence in the CR
case of the following family of operators, which includes operators of
higher orders. Set N = N∪{0}.
0
Theorem 1.2. For each (w,w′) such that n+1+w+w′ = k ∈ N and
(w,w′) ∈/ N ×N , there is a CR invariant natural differential operator
0 0
Pw,w′ : E(w,w′) → E(w−k,w′−k)
whose principal part agrees with that of ∆k.
b
Inordertoconstruct theoperatorsPw,w′, we derive atractorcalculus
for CR geometry which we anticipate will be of independent interest.
The main ingredients are to define a tractor bundle, a complex vector
bundle of rank n+2 over M, together with a CR invariant connection,
and an extension of this connection to a so-called tractor D operator
between weighted tractor bundles. We give a direct derivation of these
in terms of the pseudohermitian Tanaka-Webster connection induced
by a choice of contact form on M, parallel to the derivation in [1] in
the projective and conformal cases. The tractor bundle and connection
can also be derived from the CR Cartan connection of Cartan, Tanaka,
Chern-Moser, but we found it preferable to proceed directly, especially
POWERS OF THE SUB-LAPLACIAN 3
for the tractor D calculus needed for the construction of the invariant
operators. See [1], [9], [17], [4] for further discussion and background
about tractors. Given this machinery, the construction of the opera-
tors Pw,w′ follows a similar construction of Eastwood and Gover in the
conformal case described in [17], involving iterating tractor D’s. The
operators so constructed are strongly invariant in the sense of [9], i.e.
they act invariantly not just on scalar densities but also on density-
valued tractors. Note that the tractor construction does not yield an
operator in the case w = w′ = 0 even though the construction via the
Fefferman metric does; this is the only weight for which this occurs.
However, when n = 1, a refinement of the tractor construction does
produce a CR invariant (but not necessarily strongly invariant) oper-
ator P : E(0,0) → E(−2,−2), which by direct calculation can be
0,0
seen to agree with P . Based on the situation in the conformal case
0,0
([18]), we do not anticipate that the operators Pw,w′ and Pw,w′ agree
in general when both are defined. In fact, one must make choices of
orderings and conjugations in defining Pw,w′, and one expects that dif-
ferent choices willingeneralleadtodifferent operators. Intheflatcase,
the tractor construction (suitably interpreted) also produces operators
for all (w,w′) for which n+1 +w +w′ ∈ N; these do agree with the
corresponding operators constructed via the Fefferman metric.
A construction of invariant natural operators is given in [6] for gen-
eral parabolic geometries, of which CR geometry is a special case. This
construction produces curved analogues ofmany invariant operatorson
the flat homogeneous model, in the case when the flat operators are
standard andact between irreducible bundles corresponding to integral
non-singular highest weight vectors. For CR geometry with n ≥ 2, any
flat operator Pw,w′ satisfying these criteria has min{w,w′} ≤ −n −1.
So in these dimensions Theorem 1.2 substantially extends the results of
[6]. However, the three dimensional case is special: the relevant para-
bolic is a Borel subalgebra, so all flat operators are standard. There do
existbundlesE(w,w′)withw+w′+2 ∈ Ncorrespondingtonon-integral
or singular highest weight vectors, for which [6] does not apply. How-
ever, one can check that if (w,w′) ∈ N ×N , then the highest weight
0 0
vector for the bundle E(w,w′) is integral and non-singular. We are
not aware that it has been established in general that a given specific
operator constructed in [6] has the desired principal part, or even is
nonzero. However, it is possible to show that this is the case for all the
operators which arise in three dimensional CR geometry. Therefore, in
three dimensions Theorem 1.2 and [6] together yield the existence of a
curved version of each of the flat operators:
4 A.R.GOVER AND C. R. GRAHAM
Theorem 1.3. For each (w,w′) satisfying w +w′ +2 = k ∈ N, there
is a natural differential operator on three dimensional CR manifolds
mapping E(w,w′) → E(w−k,w′−k), whose principal part agrees with
that of ∆k.
b
Kengo Hirachi has informed us that he has established the existence of
such an operator in three dimensions mapping E(1,1) → E(−3,−3) via
the CR ambient metric with ambiguity derived in [26]. We are grateful
to Andi C˘ap for pointing out to us the special applicability of [6] in the
three dimensional case.
In Section 2 we review the basic facts about CR structures and pseu-
dohermitian geometry. We derive the tractor bundle, its connection,
curvature, and the tractor D operators in Section 3. These in turn are
used in Section 4 to construct the operators Pw,w′. The main point
in proving Theorem 1.2 is that there are natural candidate operators
which can be written down in terms of iterated tractor D operators
whose principal part is a multiple of ∆k and it must be determined
b
when the multiple is nonzero. This leads to some numerology in cal-
culating the multiple for flat space. Finally in Section 5 we review
Lee’s formulation of the Fefferman metric and the GJMS construction
of the conformal operators and present the details of the proof of Theo-
rem 1.1. We show how this derivation via the Fefferman metric can be
reformulated in terms of Fefferman’s ambient metric and also in terms
of the K¨ahler-Einstein metric of Cheng-Yau when M is a hypersurface
in Cn+1, and close with a brief discussion of Q-curvature in the CR
case. Our presentation in this section in terms of the Fefferman metric
is influenced by the point of view explicated in [14]. In Sections 4 and
5 we also prove that the operators Pw,w′ and Pw,w′ are self-adjoint (for
Pw,w′ the orderings of the D operators in the iteration must be chosen
appropriately).
It is a pleasure to thank Andi C˘ap, Mike Eastwood and Kengo
Hirachi for helpful discussions.
2. Pseudohermitian Geometry
A CR-structure on a (2n+1)-dimensional smooth manifold M is an
n-dimensional complex subbundle T1,0 ⊂ CTM such that T1,0∩T1,0 =
{0} and satisfying the integrability condition [T1,0,T1,0] ⊆ T1,0, where
we have used the same notation T1,0 for the bundle and the space
of its smooth sections. Set T0,1 = T1,0 and define Λ1,0 ⊂ CT∗M by
Λ1,0 = (T0,1)⊥. The canonical bundle K := Λn+1(Λ1,0) is a complex
line bundle on M. We will assume that K admits an (n + 2)nd root
and we fix a bundle denoted E(1,0) which is a −1/(n + 2) power of
POWERS OF THE SUB-LAPLACIAN 5
K. The bundle E(w,w′) := (E(1,0))w ⊗(E(1,0))w′ of (w,w′)-densities
is defined for w,w′ ∈ C satisfying w − w′ ∈ Z. (If E(1,0) \ {0} is
viewed as a C∗-principal bundle, then E(w,w′) is the bundle induced
by the representation λ → λwλw′ of C∗.) Without further comment,
we will assume implicitly whenever we write E(w,w′) that w,w′ ∈ C
and w −w′ ∈ Z. We will also usually write E(w,w′) for the space of
sections of this bundle; the interpretation should be clear from context.
The bundle
H = Re(T1,0)
is a real 2n-dimensional subbundle of TM. H carries a natural almost
complex structure map given by J(Z + Z¯) = i(Z − Z¯) for Z ∈ T1,0,
which induces an orientation on H. We will assume that M is ori-
entable, which implies that the bundle H⊥ ⊂ T∗M admits a nonvan-
ishing global section. A pseudohermitian structure on M is the choice
of such a contact 1-form θ. We fix an orientation on the bundle H⊥
and restrict consideration to θ’s which are positive relative to this ori-
entation. The Levi-form of θ is the Hermitian form h on T1,0 defined
by
h(Z,W) = −2idθ(Z,W).
We assume that the CR structure is nondegenerate, i.e. h is a nonde-
generate form, whose signature we denote by (p,q), p+q = n. Given
a pseudohermitian form θ, define T to be the unique vector field on M
satisfying
(2.1) θ(T) = 1 and T dθ = 0.
An admissible coframe is a set of (1,0) forms {θα}, α = 1,··· ,n, which
satisfy θα(T) = 0 and whose restrictions toT1,0 forma basis for (T1,0)∗.
We will use lower case Greek indices to refer to frames for T1,0 or its
dual. We may also interpret these indices abstractly, so will denote by
Eα the bundle T1,0 (or its space of sections) and by E its dual, and
α
similarly for the conjugate bundles or for tensor products thereof. By
integrability and (2.1), we have
dθ = ih θα ∧θβ
αβ
for a smoothly varying Hermitian matrix h , which we may interpret
αβ
as the matrix of the Levi form in the frame θα, or as the Levi form
itself in abstract index notation.
It is shown in Lemma 3.2 of [30] (see also (4.3) of [31]) that if ζ is
a (locally defined) nonvanishing section of K, then there is a unique
pseudohermitian form θ with respect to which ζ is volume normalized
6 A.R.GOVER AND C. R. GRAHAM
in the sense that
θ∧(dθ)n = in2n!(−1)qθ ∧(T ζ)∧(T ζ).
This is equivalent to the requirement that ζ have unit length with re-
spect to the natural norm induced by θ and h. Upon replacing ζ by λζ,
whereλisasmoothC∗-valuedfunction, θ isreplaced by|λ|2/n+2θ. Now
−1/(n+2) −1/n+2
|ζ|−2/n+2 := ζ−1/(n+2) ⊗ ζ is a section of K−1/n+2 ⊗K =
E(1,1), so θ := θ ⊗|ζ|−2/n+2 defines a section of T∗M ⊗E(1,1) which
depends only on the CR structure. The Levi form of θ scales the same
way, so defines a canonical section h of E (1,1), where in general
αβ αβ
we denote the tensor product with a density bundle by omitting the
second E: E (w,w′) := E ⊗E(w,w′). We will use exclusively h and
α α αβ
its inverse hαβ ∈ Eαβ(−1,−1) to raise and lower indices, so that raising
and lowering indices changes weight. The choice of pseudohermitian
form θ on M is equivalent to the choice of a CR-scale 0 < t ∈ E(1,1)
related by θ = tθ, and we also have h = th . If Υ ∈ C∞(M), the
αβ αβ
change of pseudohermitian structure θˆ = eΥθ results in hˆ = eΥh
αβ αβ
and tˆ = e−Υt. A pseudohermitian structure is said to be flat if it is
locally equivalent to the Heisenberg group with its standard pseudo-
hermitian structure. It is said to be CR flat if there is a rescaling of θ
which is pseudohermitian flat.
A choice of pseudohermitian structure determines a connection on
TM, the Tanaka-Webster, or pseudohermitian, connection ([33], [34]).
It is given in terms of an admissible coframe by
(2.2) ∇θα = −ω α ⊗θβ, ∇θ = 0,
β
where Webster’s connection 1-forms ω α satisfy
β
dθα = θβ ∧ω α +Aα θ ∧θβ,
β β
and the pseudohermitian torsion tensor A = A ∈ E is sym-
αβ αβ (αβ)
metric. (According to our conventions for raising indices, Aα is a
β
section of Eα (−1,−1).) In particular, this connection preserves T1,0,
β
so induces connections on Eα and E . One has ∇h = 0. There is an
α
induced connection on the canonical bundle, and therefore also on all
the density bundles E(w,w′).
Proposition 2.1. ∇θ = 0
Proof. Thedefinition isθ = θ⊗|ζ|−2/n+2, where ζ isvolume normalized
with respect to θ. Since ∇θ = 0, it suffices to show that ∇|ζ|2 = 0 for
suchζ. Chooseanadmissiblecoframe{θα}suchthatζ = θ∧θ1∧···∧θn.
POWERS OF THE SUB-LAPLACIAN 7
The condition that ζ is volume normalized is equivalent to det(h ) =
αβ
(−1)q. By (2.2) we have
(2.3) ∇ζ = −ω α ⊗ζ.
α
It follows that ∇|ζ|2 = −(ω α + ω α) ⊗ |ζ|2. However, from ∇h = 0
α α
one obtains ω α +ω α = hαβdh = 0 as desired. (cid:3)
α α αβ
From Proposition 2.1 we conclude that ∇t = 0, where t is the scale
associated to θ, and that ∇h = 0.
The pseudohermitian connection preserves the splitting CTM =
T1,0 ⊕ T0,1 ⊕ spanT. Therefore, if we decompose a tensor field rela-
tive to this splitting (and/or its dual), we may calculate the covariant
derivative componentwise. Each of the components may be regarded
as a section of a tensor product of Eα or its dual or conjugates thereof.
Therefore we will often restrict consideration to the action of the con-
nection on Eα or E . We will use indices α,α,0 for components with
α
respect to the frame {θα,θα,θ} and its dual, so that the 0-components
incorporate weights. If f is a (possibly density-valued) tensor field, we
will denote components of the (tensorial) iterated covariant derivatives
of f in such a frame by preceding ∇’s, e.g. ∇ ∇ ···∇ f. As usual,
α 0 β
such indices may alternatelybeinterpreted abstractly. So, forexample,
if f ∈ E (w,w′), we will consider ∇f as the triple ∇ f ∈ E (w,w′),
β β α β αβ
∇ f ∈ E (w,w′), ∇ f ∈ E (w−1,w′−1).
α β αβ 0 β β
The torsion and curvature of ∇ can be described by structure equa-
tions for the forms ω β (see (2.2) above and (1.3) of [29]), or by com-
α
muting second derivatives on functions and 1-forms (see Lemma 2.3 of
[31]. We generally follow the same conventions as [31], [29], except that
we substitute θ,hfor θ,hso thatall quantities arenaturallyweighted.)
These can be expressed in terms of the pseudohermitian curvature ten-
sor R ∈ E (1,1), A , ∇ A , and conjugates of the latter two.
αβγδ αβγδ αβ γ αβ
Flat pseudohermitian structures are characterized by the vanishing of
R and A . The Webster-Ricci tensor is defined by
αβγδ αβ
R = R γ ∈ E
αβ γ αβ αβ
and the Webster scalar curvature by
R = R α ∈ E(−1,−1).
α
From these we define
1 1
P := R − Rh .
αβ n+2 (cid:18) αβ 2(n+1) αβ(cid:19)
We will also need to know how to commute derivatives of densities.
8 A.R.GOVER AND C. R. GRAHAM
Proposition 2.2. If f ∈ E(w,w′), then
∇ ∇ f −∇ ∇ f = 0
α β β α
w −w′
∇ ∇ f −∇ ∇ f = R f −ih ∇ f
(2.4) α β β α n+2 αβ αβ 0
w −w′
∇ ∇ f −∇ ∇ f = (∇γA )f +A ∇γf.
α 0 0 α γα αγ
n+2
Proof. We first note that (2.4) holds when w = w′ = 0, since in that
case it agrees with Lemma 2.3 of [31]. Now it is a straightforward
consequence of the definitions that if V is a vector bundle with con-
nection over a manifold M, which itself has a linear connection, if vA
is a section of V, then
(2.5) [∇ ,∇ ]vA = Ω AvB −Tk∇ vA,
i j ijB ij k
where here i,j,k label TM, Ω denotes the curvature of the connection
on V, Tk denotes the torsion of the connection on TM, and the second
ij
covariantderivativeiswithrespecttothecoupledconnectiononT∗M⊗
V. We first apply (2.5) taking V to be the trivial bundle with the
flat connection over our pseudohermitian manifold with its connection:
the case w = w′ = 0. The curvature term vanishes, and we conclude
that the torsion term in (2.5) is exactly the right hand side of (2.4) for
w = w′ = 0. Next letw = −(n+2),w′ = 0, andtake V = E(w,w′) = K
with its induced connection. The form of the torsion term in (2.5) is
independent of the choice of V, so takes exactly the same form as in
the previous case. To identify the curvature term, observe that since
K is a line bundle, its curvature is a scalar 2-form on M. Choose an
admissible coframe and set ζ = θ ∧θ1 ∧···∧θn. Then we have (2.3),
so the connection form for K relative to the frame {ζ} is −ω α. The
α
curvature of K is therefore −2dω α. However, (2.3), (2.4) of [31] state
α
that
dω α = R θα ∧θβ +∇βA θα ∧θ −∇βA θα ∧θ.
α αβ βα βα
Thus (2.5) reduces to (2.4) for w = −(n+2),w′ = 0. The general case
(cid:3)
now follows upon conjugating, taking powers, and adding.
Proposition 2.2 together with the formulae in Lemma 2.3 of [31] enable
one to calculate the effect of commuting covariant derivatives on any
weighted tensor field.
If θˆ= eΥθ is another pseudohermitian form, we can express the con-
ˆ
nection, curvature, and torsion forθ in terms of that forθ. We consider
first the relation between the induced connections on tensor products
of Eα or its dual or conjugates thereof. For efficiency in expressing the
POWERS OF THE SUB-LAPLACIAN 9
transformation laws, we will denote covariant derivatives of Υ with in-
dices: ∇ ∇ Υ = Υ . One observes first that if {θα} is an admissible
α β βα
coframe for θ, then {θˆα = θα + iΥαθ} defines an admissible coframe
for θˆ. Since the restrictions of θˆα and θα to T1,0 agree, the components
of a section f of Eα or its dual (or conjugates or tensor products) are
the same in the two frames. We will denote by ∇ f, ∇ f, and ∇ f
α α 0
the components of ∇f relative to {θˆα,θˆα,θ}. For example, if f is an
b b b
unweighted function, one obtains upon changing frames that
b
(2.6) ∇ f = ∇ f, ∇ f = ∇ f, ∇ f = ∇ f+iΥγ∇ f−iΥγ∇ f.
α α α α 0 0 γ γ
The cobnnection formbs ωˆ β for ∇ cabn be expressed in terms of ω β
α α
and Υ; this is Lemma 3.4 of [29]. It is straightforward to use this to
b
calculate the analogue of (2.6) for a section τ of E . One obtains:
β β
∇ τ = ∇ τ −Υ τ −Υ τ
α β α β β α α β
(2.7) ∇ τ = ∇ τ +h Υγτ
bα¯ β α¯ β βα¯ γ
∇ τ = ∇ τ +iΥγ¯∇ τ −iΥγ∇ τ −i(Υγ −ΥγΥ )τ .
b0 β 0 β γ¯ β γ β β β γ
We alsobneed to know how the connection transforms on densities.
Proposition 2.3. If f ∈ E(w,w′), then
∇ f =∇ f +wΥ f
α α α
∇b f =∇ f +w′Υ f
α α α
∇b f =∇ f +iΥγ∇ f −iΥγ∇ f
0 0 γ γ
b + n+12 (w+w′)Υ0 +iwΥγγ −iw′Υγγ +i(w′ −w)ΥγΥγ f
(cid:2) (cid:3)
Proof. As in the proof of Proposition 2.2, if we establish the result
for w = −(n + 2),w′ = 0, the general case follows by conjugating,
taking powers, and adding. Let ζ = θ ∧ θ1 ∧ ···∧ θn be a section of
K = E(−(n + 2),0). Then ∇ζ is given by (2.3). However, also ζ =
e−Υθˆ∧θˆ1∧···∧θˆn, so applying (2.3) again gives ∇ζ = −(ωˆ α+dΥ)⊗ζ.
α
Therefore
b
∇ζ −∇ζ = −(ωˆ α −ω α +dΥ)⊗ζ.
α α
Now Lemma 3.4 of [29] gives ωˆ β − ω β in terms of Υ. Contracting,
b α α
adding dΥ, and reformulating the result in terms of components yields
(cid:3)
the desired formulae.
The curvature and torsion also transform under the pseudohermitian
change. From Lemma 2.4 of [31] one obtains
(2.8) P = P − 1(Υ +Υ )− 1Υ Υγh
αβ αβ 2 αβ βα 2 γ αβ
b
10 A.R.GOVER AND C. R. GRAHAM
and
(2.9) A = A +iΥ −iΥ Υ .
αβ αβ αβ α β
We also will need the transformation laws for two other objects. Set
b
P = P α = 1 R and define
α 2(n+1)
1
T = (∇ P −i∇βA ) ∈ E (−1,−1),
α α αβ α
n+2
1
S = − (∇αT +∇α¯T +P Pαβ −A Aαβ) ∈ E(−2,−2).
n α α¯ αβ αβ
Straightforwardcalculation using the formulae discussed above and the
Bianchi identities of Lemma 2.2 of [31] gives:
T = T + iΥ +P βΥ −iA Υβ
α α 2 0α α β αβ
+ 1Υ Υβ − 1Υ βΥ − 1Υ ΥβΥ ,
b 2 αβ 2 α β 2 β α
S = S + 1Υ −3(ΥαT +ΥαT )
2 00 α α
b +i(Υ0αΥα −Υ0αΥα)− 14(Υ0)2 + 32i(AαβΥαΥβ −AαβΥαΥβ)
−3P ΥαΥβ − 1(Υ ΥαΥβ +Υ ΥαΥβ)
αβ 2 αβ αβ
+ 1(Υ +Υ )ΥαΥβ + 3(Υ Υα)2.
2 αβ βα 4 α
We derived the expressions for T and S from the condition that they
α
satisfy transformation laws of this form, which we needed to construct
the tractor connection in the next section. Only later did we observe
that they occur already in [30] as components of the connection forms
and Ricci tensor of the Fefferman metric. Also, T and a variant of S
α
arise in [29], where they were combined with P and A to form a 2-
αβ αβ
tensor on M which was used to construct a version of pseudohermitian
normal coordinates. In that work, only the part of the transforma-
tion laws involving highest derivatives (counted non-isotropically) of Υ
was relevant. The version of S in [29] therefore does not include the
P Pαβ −A Aαβ term, which however is important for us.
αβ αβ
3. Tractors
The pseudohermitian connection depends on the choice of θ. In this
section we construct a vector bundle of rank n+2 over M, the tractor
bundle, together with a CR invariant tractor connection. We also show
how to extend this connection to a CR invariant tractor D operator on
weighted tractors which can be iterated.
For a given choice of θ, the (co-)tractor bundle E is realized as a
A
direct sum
(3.1) E = E(1,0)⊕E (1,0)⊕E(0,−1).
A α