by Tom Branson and Rod Gover

*Curvature prescription*

Everything below will take place in the setting of Riemannian manifolds (or Riemannian conformal manifolds) of even dimension . Of course many statements will also be true for odd-dimensional manifolds and/or pseudo-Riemannian (conformal) manifolds, but our main intent is to make this blurb readable. There will be no reference list here, though there are plans to compile a separate reading list (of real papers) on the topic.

A touchstone in Differential Geometry is the *Yamabe equation*:
for ,

the given by () is the scalar curvature of .

**Exercise 1**.
Show that () implies the conformal change law
for the *conformal Laplacian*

namely

Here, and in such formulas below, the function on the very right is to be viewed as a multiplication operator, so the relation really says that for all smooth functions ,

When , the equation governing the conformal change of is
qualitatively different from a PDE standpoint:

There is a formal procedure of analytic continuation in dimension
(which in fact can be made rigorous) that allows one to guess
(or prove) ( given ()). The Yamabe equation may be
rewritten as

Note that we have slipped in an extra on the left. The advantage of this is that, as a power series in , all terms in the equation begin at the first power. Dividing by and then evaluating at , we get ().

**Exercise 2** (following C.R. Graham).
Make the dimensional continuation
argument rigorous by looking at *stabilisations* of the manifold ,
i.e. the -dimensional manifolds , where is the
standard -torus.

There is a generalisation of this whole picture to higher order, in which the role of the pair is played by a pair consisting of an operator and a local scalar invariant. The are the celebrated Graham-Jenne-Mason-Sparling (GJMS) operators, which by construction have the following properties.

- exists for even and .
- . (Here and below, LOT``lower order terms''.)
- is formally self-adjoint.
- is conformally invariant in the sense that

- has a polynomial expression in and in which all coefficients are rational in the dimension .
- has the form

where is a local scalar invariant, and is an operator on 1-forms of the form

With these properties, conditions are right to generalise the Yamabe
equation to

where

Analytic continuation in dimension then yields the following analogue of the Gauss curvature prescription equation. If we denote and simply by and , then

Though there is much more to be said about the -curvature, this is probably the central formula of the theory.

**Exercise 3**. Show that if we have a local invariant satisfying
a conformal change law like (,
, with
a natural differential operator, then necessarily is conformally
invariant in the sense
.

The fact that has an expression with rational dependence on the dimension is crucial to making the analytic continuation rigorous, whether one does it by stablilisation (generalising Exercise 2), or by an algebraic argument using -linear combinations in a dimension-stable basis of invariants.

Very explicit formulas for and are
known up to . The case, which was already being discussed
in the early 1980's, is particularly appealing as a source of intuition,
since the formulas there are still quite manageable. Let be the
Ricci tensor and let

The

where

and

Here is the natural action of a symmetric 2-tensor on 1-forms.

The transition from to already points up the fact that the are not uniquely determined: if is the Weyl conformal curvature tensor, we could add a suitable multiple of to without destroying any of its defining properties. (The coefficient should be rational in , should have a zero at , and should not have poles that create new ``bad'' dimensions.)

The situation is also already big enough to show that the study
of is not just a disguised
study of the conformal properties of the Pfaffian
. One of the salient properties of the Pfaffian is that
it can be written as a polynomial in , without any explicit
occurrences of . For example, in dimension 4,

But the term in is an absolutely essential aspect, and it generalises: see Exercise 6 below.

All the conformal change laws we've mentioned are good in odd dimensions,
and that for in dimension 3 shows that very strange nonlinearities
can occur:

where .

Back to the general case, a celebrated property of is the conformal
invariance of its integral on compact manifolds:

Indeed, since ,

But is an exact divergence, by the property of , and thus it integrates to 0, showing invariance. This has an immediate generalisation. If is a smooth function,

Since is formally self-adjoint, we may move it over to in the very last term under the integral. If it happens that , there is no contribution from this term. Thus

*Relativistic considerations*

The Einstein equations are obtained by taking the *Einstein-Hilbert
action* in dimension 4, and taking the *total metric
variation*. This means we take a compactly supported
symmetric tensor and a curve
of metrics with
, and compute that

where is any compact set containing supp. Here we may view as the pairing of a covariant with a contravariant symmetric tensor, or as the metric () pairing of two covariant tensors.

*Weyl relativity* is one proposal for replacing the Einstein-Hilbert
action with an action that is invariant under multiplication of the
metric by a positive constant: under the variation above,

where is the

One aspect of is that the nonlinear differential operator

carrying a metric to its Bach tensor, is fourth-order quasilinear. Its linearisation is an interesting fourth-order conformally invariant linear differential operator.

In attempting to generalise this to higher dimensions, it's clear that won't help - its total metric variation is 0, since it's a topological invariant. Choices like for are uninteresting because the linearisations of the analogues of the operators () have order lower than one might hope for - less than . In fact these linearisations will even vanish when we vary at a conformally flat metric.

Coming to the rescue of the situation is :

where is the

**Exercise 4**. Show that if is conformally invariant,
then its total metric variation tensor is conformally invariant.
Show that if is any conformally invariant tensor, then
the linearisation of the map is conformally invariant
on trace-free perturbations of (and 0 on pure trace perturbations).

*Quantum considerations*

Let be a natural differential operator with positive definite
leading symbol, and suppose is a positive power of a conformally
invariant operator. For example, could be one of the GJMS
operators, or it could be the square of the Dirac operator.
Then in dimensions 2,4,6, and conjecturally in higher even
dimensions,

Such formulas are the finite variational formulas corresponding to
*Polyakov formulas*, which are infinitesimal variational formulas
for the determinant; these take the form

In fact, getting from ( to ())
may be viewed as a process of finding *conformal primitives*.
We say that a functional on the conformal class
is a conformal primitive for a local invariant if
. Of course this should happen at all possible
choices of background metric , and all directions of variation
. This can be said in a more invariant way, following
a suggestion of Mike Eastwood. Putting the
``running'' metric and the background metric on the same footing
in a two-metric functional
on the conformal class, we
require of a conformal primitive that it be

- alternating and in the and arguments;
- cocyclic in the sense that

- having variation in the sense above when varied in for fixed .

Some local invariants have other local invariants as
conformal primitives. For example, since

we have

so

This makes a conformal primitive for . is an example of a local invariant that does not have such a local conformal primitive.

In order to handle these objects more cleanly,
let's view and as being density-valued objects and , so
that a ``weight term'' involving the conformal factor does not appear
explicitly. In other words, replace by
, and
by
.
(For readers unfamiliar with densities, not much is lost conceptually
in assuming our manifold is oriented and talking about -forms instead
of scalar densities.)
Then

Recall that when we wrote this, we were thinking about a background metric
and a perturbed metric .
But there
is an interesting way of rewriting the first term,
as

and clearly

The functional () is

Since the log-determinant functional will obviously satisfy the cocycle condition (), and since the second functional in () satisfies such a condition, we expect to behave similarly. One way to see that this expectation is fulfilled is to use the conformal primitive property: for fixed and , with for above, the two sides of () have the same conformal variation (of ), and the same value at .

**Exercise 5**. Show that if ,
,
, and
are 4 conformally
related metrics, then

The following conjecture would be enough to prove the conjecture mentioned at the beginning of the section on the form of the determinant quotient.

**Conjecture 1**. If is a natural -form and is
conformally invariant, then

where is a local conformal invariant and has a local conformal primitive. That is, there is a local invariant for which the conformal variation of is .

The point of separating these 3 kinds of terms is that
will
have a very banal conformal primitive, namely itself, or (to write it
in a way that makes the properties of a conformal primitive more apparent),

has an interesting conformal primitive, as discussed above. is not uniquely defined, but the The in the statement of the conjecture could be anyone's favorite version of . In fact, is well-defined up to addition of an .

Then there are the following related conjectures:

**Conjecture 2**. Any as above may be written

where is an exact divergence.

**Conjecture 3**. Any as above may be written

There are at least 2 filtrations of the local invariants of this type that
should be relevant. First, any invariant can be written as a sum of
monomial expressions in and with
, where
(resp. ) is the number of occurrences of (resp. )
in the monomial. If an invariant can be written with
for each monomial term, let's say
. Then

**Exercise 6**. Use the conformal change law for to show that
the class of in
is nontrivial, and agrees with the class of
.
This establishes that
*Pff* and are ``at opposite ends'' of the -filtration.

The other filtration is by the degree of the conformal change law.
If

with of homogeneity with respect to scalar multiples of , then say if for . Then local conformal invariants are in , and is in .

*Other routes to and its variants*

There is an alternative definition of which avoids dimensional
continuation. We
write for the space of smooth functions,
for space of smooth
1-forms and define the special section

of the direct sum bundle . Let us first set the dimension to be 4, simply present the some results and then explain how this works. Then we get

where is the coupled conformal Laplacian operator. More precisely , which appears to be the usual formula for the conformal Laplacian (cf. above), but now is a connection which couples the usual metric connection with the connection

on the sum bundle . This bundle is called the standard tractor bundle and this connection is usually termed the (normal conformal) tractor connection. It is equivalent to a principal bundle structure known as the (normal conformal) Cartan connection. For those who know about Cartan connections we can say that the tractor bundle and connection is an associated bundle and connection for the Cartan bundle. We have used a metric to express these objects in terms of a Riemannian structure but in fact the bundle and connection are conformally invariant and so descend to well defined structures on a conformal manifold. In fact, to be more accurate, the decomposition of the standard tractor bundle is really where indicates the space of conformal densities of weight . The field is a section of . In this section and the next we are allowing tensors and tractor fields to be density valued, to simplify the notation, but partially suppressing the details of weights involved.

This construction generalises. In each even dimension there is a
conformally invariant differential operator
so that
for any metric we have

where is a well known second order conformally invariant linear differential operator known as the tractor operator. From this and ( it follows that the -curvature , for , differs from by a linear conformally invariant operator acting on . In fact it follows easily from the definition of that

where is the GJMS operator of order . So we have recovered the now famous property (cf. ()).

As a final comment on the above story we should clarify the origins of
the tractor field
defined and used above. For those who are
familiar with tractors a more enlightening alternative definition is

Here is the tractor operator and is the so-called fundamental operator. is the conformal scale corresponding to the metric . The point is that these operators are both conformally invariant and under we have . Since satisfies a Leibniz rule and is a ``logarithmic derivative'' the conformal transformation law of is no surprise.

The main results above are derived via the ambient metric construction of Fefferman and Graham. Explaining this construction would be a significant detour at this point. Suffice to say that this construction geometrically associates to an -dimensional conformal manifold an -dimensional pseudo-Riemannian manifold . The GJMS operators arise from powers of the Laplacian , of , acting on suitably homogeneous functions. The operators arise in a similar way from on appropriately homogeneous sections of the tangent bundle . Such homogeneous sections correspond to tractor fields on the conformal manifold . The results above are given by an easy calculation on the ambient manifold. Thus we can take () as a definition of the -curvature; it is simply the natural scalar field that turns up on the right hand side.

While this definition avoids dimensional continuation, there is still the issue of
getting a formula for . There is an effective algorithm for
re-expressing the ambient results in terms of tractors which then
expand easily into formulae in terms of the underlying Riemannian
curvature and its covariant derivatives. This solves the problem for small .
For example

While such formulae shed some light on the nature of the -curvature it would clearly be ideal to give a general formula or simple inductive formula. From the angle discussed here the missing information is a general formula for the operators .

**Problem 1:** Give general formulae or inductive formulae
for the operators
.

This seems to be a difficult problem. In another direction there is
another exercise to which we already have some answers. One of the
features of the -curvature is that it ``transforms by a linear operator''
within a conformal class.
More precisely, it is an example of a natural Riemannian tensor-density field
with a transformation law

**Problem 2:** Construct other natural tensor-densities which transform according
to (). (Note that any solution yields a conformally invariant natural operator .)

From the transformation law for above, we can evidently
manufacture solutions to this problem. We have observed already that
is a section of the bundle
. If
is any scalar (or rather density) valued natural conformally invariant
differential operator which acts on
then can act on
, and has a conformal transformation of the the form
(). Using the calculus naturally associated to
tractor bundles (or equally effectively,
using the ambient metric) it is in fact a
simple matter to write down examples, and the possibilities increase with
dimension.
This is most interesting when the resulting scalar field
gives a possible modification to the original -curvature. For those
familiar with densities this means that should take values in
densities of weight ; this is the weight
at which densities that can be integrated
on a conformal manifold. For example, in any dimension we
may take to be
where indicates a contracted action of the tractor
operator and the square of the Weyl curvature is here viewed as a
multiplication operator. In dimension this takes values in
and we have

and

Note that in this example the conformally invariant ``-operator'' is formally self-adjoint. So for any constant , is another scalar field with almost the same properties as . It is not so closely related to the GJMS operator , but it is related instead to a modification of by . It is clear that solutions to Problem 2 have a role to play in the problem of characterising the -curvature and the GJMS operators.

*A generalisation: maps like *

So far we have viewed the -curvature as a natural scalar field. It turns out that if instead we view it as an operator then it fits naturally into a bigger picture. To simplify matters suppose we are working with a compact, oriented, but not necessarily connected, manifold of even dimension . We fix and so omit in the notation for . We can view as a multiplication operator from the closed 0-forms (i.e. the locally constant functions) into the space of -forms (which we identify with via the conformal Hodge ). With the observations above we have the following properties:

- is not conformally invariant but , where is a formally self-adjoint operator from 0-forms to -forms. has the form which implies the next properties.
- is conformally invariant and non-trivial in general.
- If and then is conformally invariant.
- (See the discussion immediately below.) In each choice of metric is formally self-adjoint.
- (See the discussion immediately below.) is the -curvature.

The idea now is to look for analogous operators on other forms. We
write
for the space of closed -forms. Consider the operator

**Exercise 7**.
On
we have

where is some nonzero constant, , and in the display is viewed as a multiplication operator.

Note that the conformal variation term is the
Maxwell operator and is formally self-adjoint. So satisfies the
analogue of property 1 above. The analogue of property 3 is an
immediate consequence, i.e.,
is
conformally invariant where now is a closed
-form and
(so in fact by compactness
are both closed). Next observe, by inspection, that is formally
self-adjoint. So we have analogues for 1,3,4. There is also a bonus
property, which is clear from the transformation law displayed:

is a non-trivial conformally invariant operator. In dimension 4 this is the Paneitz operator.

So finally we need an analogue for
property 2. It is clear that is conformally invariant as a
map
,
so this is an analogue. But we can do more.
There is no reason to
suppose the image is co-closed.
On the other hand note that is conformally invariant on
and so
we have the following:

**Fact:** Let
, where , , with
the standard Riemannian structure.
Then
if and only if is harmonic.
Furthermore,
the map () is non-trivial.

In some recent work the authors have used the ambient metric, and its relationship to tractors, to show that the above construction generalises along the following lines: There are operators (), given by a uniform construction, with the following properties:

- has the conformal transformation law , where is a formally self-adjoint operator from -forms to -forms, and is a constant multiple of .
- is a conformally invariant subspace of and is conformally invariant. There are conformal manifolds on which is non-trivial.
- If
and
then

is conformally invariant. - For each choice of metric , is formally self-adjoint.
- is the -curvature.

**Problems :** There are analogues for the operators
of most of the conundrums and problems for the -curvature.

Back to the
main index
for Conformal structure in Geometry, Analysis, and Physics .