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:
Show that () implies the conformal change law
for the conformal Laplacian
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
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.
With these properties, conditions are right to generalise the Yamabe
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 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,
All the conformal change laws we've mentioned are good in odd dimensions,
and that for in dimension 3 shows that very strange nonlinearities
Back to the general case, a celebrated property of is the conformal
invariance of its integral on compact manifolds:
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
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,
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 :
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).
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
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
Some local invariants have other local invariants as
conformal primitives. For example, since
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
(For readers unfamiliar with densities, not much is lost conceptually
in assuming our manifold is oriented and talking about -forms instead
of scalar densities.)
Recall that when we wrote this, we were thinking about a background metric
and a perturbed metric .
is an interesting way of rewriting the first term,
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 ,
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
The point of separating these 3 kinds of terms is that
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),
Then there are the following related conjectures:
Conjecture 2. Any as above may be written
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
(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
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.
Other routes to and its variants
There is an alternative definition of which avoids dimensional
write for the space of smooth functions,
for space of smooth
1-forms and define the special section
This construction generalises. In each even dimension there is a
conformally invariant differential operator
for any metric we have
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
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 .
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
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
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
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:
The idea now is to look for analogous operators on other forms. We
for the space of closed -forms. Consider the 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.,
conformally invariant where now is a closed
(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:
So finally we need an analogue for
property 2. It is clear that is conformally invariant as a
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
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:
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 .