# Recent preprints and articles by A. Rod Gover

## Paper information

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: 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 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
Availability:
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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