Disclaimer: These are rough notes only, aimed at setting the scene and promoting discussion at the American Institute of Mathematics Research Conference Center Workshop `Conformal Structure in Geometry, Analysis, and Physics,' - August 2003. For simplicity, we have omitted all references. Curvature conventions are in an appendix. Conversations with Tom Branson and Rod Gover have been extremely useful.
Let be an oriented even-dimensional Riemannian -manifold. Branson's -curvature is a canonically defined -form on . It is not conformally invariant but enjoys certain natural properties with respect to conformal transformations.
When , the -curvature is a multiple of the scalar curvature. With
conventions as in the appendix . Under conformal rescaling of the
When , the -curvature is given by
For general even , the -curvature transforms as follows:-
Even when is conformally flat, the existence of is quite subtle. We can
reason as follows. When is actually flat then must vanish. Therefore,
in the conformally flat case, locally if we write
where is flat, then () implies that
That is conformally invariant on flat space is well-known. It
may also be verified directly by a rather similar calculation. For example,
here is the calculation when . For general conformally related metrics
in dimension 4,
Conundrum: Deduce fact 2 from fact 1 or vice versa. Both are consequences of (). Alternatively, construct a Lie algebraic proof of fact 2. There is a Lie algebraic proof of fact 1. It corresponds to the existence of a homomorphism between certain generalised Verma modules for .
What about a formula for , even in the conformally flat case? We have a recipe for , namely (), but it is not a formula. We may proceed as follows.
and is flat, then
() implies that
To proceed further we need two identities. If has conformal weight ,
then as described in the appendix,
The quantities in () have weight . Therefore, applying
Of course, we may continue in the vein, further differentiating () to obtain a formula for expressed in terms of complete contractions of , its hatted derivatives, and . With increasing , this gets rapidly out of hand. Moreover, it is only guaranteed to give in the conformally flat case. Indeed, when this naive derivation of fails for a general metric. Nevertheless, there are already some questions in the conformally flat case.
Conundrum: Find a formula for in the conformally flat case. Show that the procedure outlined above produces a formula for .
In fact, there is a tractor formula for the conformally flat . This is not
the place to explain the tractor calculus but, for those who know it already:-
Recall that, like , the Pfaffian is an -form canonically associated to a
Riemannian metric on an oriented manifold in even dimensions. It is defined as
a complete contraction of copies of the Riemann tensor with two copies of
the volume form. For example, in dimension four it is
Conundrum: Find a direct link between and the Pfaffian in the conformally flat case. Prove directly that is a topological invariant in this case.
Conundrum: Is it true that, on a general Riemannian manifold, may be written as a multiple of the Pfaffian plus a local conformal invariant plus a divergence?
Recall the conventions for Weyl structures as in the appendix. In particular, a
metric in the conformal class determines a -form . In fact, a
Weyl structure may be regarded as a pair
equivalence under the simultaneous replacements
structure as follows. Since is a Riemannian invariant, the differential
operator is necessarily of the form
Riemannian invariant linear differential operator from -forms to -forms.
is a Weyl structure, choose a representative metric
and consider the -form
In combination with () we obtain
Conundrum: Can we find such a in general even dimensions? Presumably, this would restrict the choice of Riemannian .
Though given by () is an invariant of the Weyl
structure, it is not manifestly so. Better is to rewrite it as follows. Using
conventions from the appendix, we may write the Schouten tensor
() of the Weyl structure in terms of the Schouten tensor of a
representative metric :-
Conundrum: Did we really need to go through this detailed calculation? What are the implications, if any, for the operator ?
Conundrum: Can we characterise the Riemannian by sufficiently many properties? Do Weyl structures help in this regard?
Tom Branson has suggested that, for two metrics and
in the same conformal class on a compact manifold , one should consider the
Conundrum: Are there any deeper properties of Branson's cocycle ?
One possible rôle for is in a curvature prescription problem:-
Conundrum: On a given manifold , can one find a metric with specified ?
One can also ask this question within a given conformal class or within the realm of conformally flat metrics though, of course, if is compact, then must be as specified by the conformal class and the topology of . There is also the question of uniqueness:-
Conundrum: When does determine the metric up to constant rescaling within a given conformal class?
Since we know how changes under conformal rescaling (), this question is equivalent to
Conundrum: When does the equation have only constant solutions?
On a compact manifold in two dimensions this is always true: harmonic functions
are constant. In four dimensions, though there are conditions under which
has only constant solutions, there are also counterexamples, even on
conformally flat manifolds. The following counterexample is due to Michael
Singer and the first author. Consider the metric in local coördinates
APPENDIX: Curvature Conventions
Firstly, our conventions for conformal weight. A density of conformal weight may be identified as a function for any metric in the conformal class. At the risk of confusion, we shall also write this function as . If however, our choice of metric is replaced by a conformally equivalent , then the function is replaced by . Quantities that are not conformally invariant can still have a conformal weight with respect to constant rescalings. For example, the scalar curvature has weight in this respect. Explicit conformal rescalings are generally suppressed.
The Riemann curvature is defined by
A Weyl structure is a conformal structure together with a choice of
torsion-free connection preserving the conformal structure. In other
words, if we choose a metric in the conformal class, then
where is the Levi-Civita connection for the metric . Let
denote the Ricci curvature of the connection :-
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