Title: Conformally invariant operators, differential forms, cohomology and a
generalisation of Q curvature
Author(s): Thomas Branson and A. Rod Gover
Status: preprint
Length: 53 pages
Math Review Classification(s): Not available
Availability:
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Abstract:
Title: Standard tractors and the conformal ambient metric construction
Author(s): Andreas Cap and A. Rod Gover
Status: preprint
Length: 26 pages
Math Review Classification(s): Not available
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Abstract:
In this paper we relate the Fefferman--Graham ambient
metric construction for conformal manifolds to the approach to
conformal geometry via the canonical Cartan connection. We show that
from any ambient metric that satisfies a weakening of the usual
normalisation condition, one can construct the conformal standard
tractor bundle and the normal standard tractor connection, which are
equivalent to the Cartan bundle and the Cartan connection. This result
is applied to obtain a procedure to get tractor formulae for all
conformal invariants that can be obtained from the ambient metric
construction. We also get information on ambient metrics which
are Ricci flat to higher order than guaranteed by the results of
Fefferman--Graham.
Title: Electromagnetism, metric deformations,
ellipticity and gauge operators on conformal 4-manifolds
Author(s): Thomas Branson and A. Rod Gover
Status: preprint
Length: 29 pages
Math Review Classification(s): Not available
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Abstract:
Title: A conformally invariant differential operator on Weyl
tensor densities
Author(s): Thomas Branson and A. Rod Gover
Status: To appear, J. Geometry and Phys.
Length: 17 pages
Math Review Classification(s): Not available
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Abstract:
We derive a tensorial
formula for a fourth-order conformally
invariant differential operator on conformal
4-manifolds. This operator is applied to algebraic Weyl tensor
densities of a certain conformal weight, and takes its values in
algebraic Weyl tensor densities of another weight. For oriented
manifolds, this operator reverses duality: For example in the
Riemannian case, it takes self-dual to anti-self-dual tensors and vice
versa. We also examine the place that this operator occupies in known
results on the classification of conformally invariant operators, and
we examine some related operators.
Title: Conformally Invariant Powers of the Laplacian, Q-curvature and Tractor Calculus
Author(s): A. Rod Gover and Lawrence J. Peterson
Status: preprint
Length: 39 pages
Math Review Classification(s): Not available
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Abstract:
Title: Invariant Theory and Calculus for Conformal Geometries
Author(s): A. Rod Gover
Status: To appear: Advances in Mathematics
Length: 52 pages
Math Review Classification(s): Not available
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Abstract:
Title: Conformally Invariant Non-Local Operators
Author(s): Thomas Branson and A. Rod Gover
Status: To Appear: Pacific Journal of Mathematics
Length: 36 pages
Math Review Classification(s): not available
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Abstract:
On a conformal manifold with boundary,
we construct conformally
invariant local boundary conditions $B$ for the conformally invariant
power of the Laplacian $\Box_k\,$, with the property that
$(\Box_k\,,B)$ is formally self-adjoint. These boundary problems are
used to construct conformally invariant
non-local operators on the boundary $\Sigma$, generalising the
conformal Dirichlet-to-Robin operator, with principal parts
which are odd powers $h$ (not necessarily positive) of
$(-\Delta_\Sigma)^{1/2}$,
where $\Delta_\Sigma$ is the boundary Laplace operator.
The constructions use tools from a conformally invariant calculus.
Title: Tractor Calculi for Parabolic Geometries
Author(s): Andreas Cap and A. Rod Gover
Status: To Appear: Transactions of the American Mathematical Society
Length: 38 pages
Math Review Classification(s): not available
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Abstract:
Parabolic geometries may be considered as curved analogues of the
homogeneous spaces $ G/P$ where $ G$ is a semi-simple Lie group and
$ P\subset G$ a parabolic subgroup. Conformal geometries and CR
geometries are examples of such structures. We present a uniform
description of a calculus, called tractor calculus, based on natural
bundles with canonical linear connections for all parabolic
geometries. It is shown that from these bundles and connections one
can recover the Cartan bundle and the Cartan connection. In
particular we
characterize the normal Cartan connection from this induced
bundle/connection perspective. We construct explicitly a family of
fundamental first order differential operators, which are
analogous to a covariant derivative, iterable and defined on all natural
vector bundles bundles on parabolic geometries. For an important
sub-class of parabolic geometries we explicitly and directly
construct the tractor bundles, their canonical linear connections
and the machinery for explicitly calculating via the tractor
calculus.
Title: Tractor Bundles for Irreducible Parabolic Geometries
Author(s): Andreas Cap and A. Rod Gover
Status: In: S.M.F. Colloques,
Seminaires & Congres 4, (2000) 129--154 .
Length: 25 pages
Math Review Classification(s):
not available
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Abstract:
We use the general
results on tractor calculi for parabolic geometries obtained in
``Tractor Calculi for Parabolic Geometries'' to give a simple and
effective characterisation of arbitrary normal tractor bundles on
manifolds equipped with an irreducible parabolic geometry (also called
almost Hermitian symmetric-- or AHS--structure in the
literature). Moreover, we also construct the corresponding normal
adjoint tractor bundle and give explicit formulae for the normal
tractor connections as well as the fundamental D--operators on such
bundles. For such structures, part of this information is equivalent
to giving the canonical Cartan connection. However it also provides
all the information necessary for building up the invariant tractor
calculus. As an application, we give a new simple construction of the
standard tractor bundle in conformal geometry, which immediately leads
to several elements of tractor calculus.
Title: Aspects of Parabolic Invariant Theory
Author(s): A. Rod Gover
Status: In: Supp.
Rend. Circ. Matem. Palermo, Ser. II, Suppl. 59, (1999) 25--47.
Length: 23 pages
Math Review Classification(s): not available
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Abstract:
These lectures include a brief discussion of parabolic geometries in
general but are concerned primarily with conformal and CR
structures. Motivated by the problems of constructing invariant
operators on tensor bundles and constructing polynomial
invariants of such structures, the lectures will describe basic
invariant operators for each of these structures which in a certain
sense are analogues of the Levi-Civita connection of Riemannian
geometry. Some applications of these to the problems mentioned
will also be treated. This work was presented as a series of three
lectures at the $18^{\rm th}$ Winter School on Geometry and Physics,
Srn\'{\i}, Czech Republic, January 1998.
Title: Local Twistor Calculus for Quaternionic Structures and
Related Geometries
Author(s): A.R. Gover and J. Slovak
Status: In: Journal of Geometry and Physics, 32 (1999) 14--56.
Length: 42 pages
Math Review Classification(s): not available
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Abstract:
New universal invariant operators are introduced in a class of
geometries which include the quaternionic structures and their
generalisations as well as 4-dimensional conformal (spin) geometries.
It is shown that, in a broad sense, all invariants and invariant
operators arise from these universal operators and that they may be
used to reduce all invariants
problems to corresponding algebraic problems involving homomorphisms
between modules of certain parabolic subgroups of Lie groups. Explicit
application of the operators is illustrated by the construction of all
non-standard operators between exterior forms on a large class of the
geometries which includes the quaternionic structures.
Title: The Funk Transform as a Penrose
Transform
Author(s): T.N. Bailey, M.G. Eastwood, A. R. Gover and L.J. Mason
Status: In: Mathematical Proceedings of the Cambridge Philosophical
Society, 125, (1999) 67--81.
Length: 14 pages
Math Review Classification(s): not available
Availability:
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Abstract:
The Funk transform is the integral transform from the space of smooth even
functions on the unit sphere $S^2 \subset \Bbb R^3$ to itself defined by
integration over great circles. One can regard this transform as a limit
in a certain sense
of the Penrose transform from $\CP$ to $\CPs$. We exploit this viewpoint
by developing a new proof of the bijectivity of the Funk transform which
proceeds by considering the cohomology of a certain involutive (or formally
integrable) structure on an intermediate space.
This is the simplest example of what we hope will prove to be a general
method of obtaining results in real integral geometry by means of complex
holomorphic methods derived from the Penrose transform.
Last updated: 16 September 2001