A method constructs tractor conformal bundles for spacelike submanifolds in Lorentzian manifolds.
problem Characterize conditions for a tractor conformal bundle to be standard and normal.
method Introduce a canonical construction of a tractor conformal bundle and characterize conditions for it to be standard and normal.
result Characterizes conditions for a tractor conformal bundle to be standard and normal.
Embedding theorem for tractor bundles applied to conformal geometry.
problem Embedding theorem for tractor bundles in Cartan geometries.
method Extension of Gromov-Zimmer embedding theorem to tractor bundles.
result Rigidity result for conformal actions of special pseudo-unitary groups.
Study parallel tractors and cotractors on almost Grassmannian structures.
problem Characterize parallel tractors and cotractors on almost Grassmannian structures.
method Provide explicit formulae for splitting operators, first BGG operators, and prolongation connections. Characterize solutions of the BGG operators geometrically.
result Describe the geometry of the zero locus of solutions of the first BGG operators.
CR Killing operator derived from tractor calculus for CR structures.
problem Analyzing CR structures and their deformations.
method Tractor calculus and BGG operators applied to compatible almost CR structures.
result CR Killing operator is a first BGG operator for the modified adjoint tractor connection.
New tractor geometry derived from asymptotically flat spacetimes.
problem Understanding the geometry of spacetimes near their boundaries.
method Derived null-tractor bundle from interior spacetime geometry, proved connections' uniqueness, and expressed results in BMS coordinates.
result Tractor connection encodes mass and angular momentum in 3D, and asymptotic shear in higher dimensions.
The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor connections, the various structures they can preserve, and their geometric interpretations…
The paper studies conformal and projective structures using 2-frame bundles.
problem Understanding connections between conformal and projective structures.
method Using the dressing field method to obtain local, gauge invariant connections.
result A projective tractor bundle can be defined using the same construction as for conformal structures.
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 tract…
The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing rep…
New geometrization of gravitational wave phase space.
problem Understanding the geometry of null-infinity in asymptotically flat space-times.
method Proposes a new geometrization using tractor calculus adapted to degenerate conformal metrics.
result Gravitational waves correspond to a class of tractor connections called 'null-normal'.
We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the …
There is a well known one--parameter family of left invariant CR structures on SU(2)≅S3. We show how purely algebraic methods can be used to explicitly compute the canonical Cartan connections associated to these structures and their curvatures. We also obtain explicit descriptions of tractor bundles and tracto…
Compact formulas for Yang-Mills conditions on conformal manifolds.
problem Finding variational analogs of Yang-Mills conditions.
method Provided compact formulas and proved their equivalence to Yang-Mills conditions under specific metrics.
result Conformal Yang-Mills condition is equivalent to vanishing of Fefferman-Graham obstruction tensor.
Quantizes contact structures using dynamical methods.
problem Quantizing contact structures in a flat connection.
method Constructs a dynamical quantization using a flat connection on a Hilbert tractor bundle.
result Determines a contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics.
Develops a new calculus for contact structures on manifolds.
problem No specific problem stated; focuses on developing calculus.
method Elementary construction of invariant partial connection on quotient bundle.
result Constructs a second order invariant differential operator.
Establishes a connection between Kähler metrics and vector bundle sections.
problem Finding Kähler metrics in a conformal class.
method One-to-one correspondence between Kähler metrics and parallel sections of a vector bundle with conformally invariant connection.
result Obstructions for a Riemannian metric to be conformal to a Kähler metric.
Defines a new energy for submanifolds, comparing to Willmore energy.
problem Finding new conformally invariant energies for submanifolds.
method Coupling tractor connection to GJMS operators for higher-dimensional analogues.
result Shows comparison with existing energy in 4D.
The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a `projective cone', a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying…
This paper aims to classify the holonomy of the conformal Tractor connection, and relate these holonomies to the geometry of the underlying manifold. The conformally Einstein case is dealt with through the construction of metric cones, whose Riemmanian holonomy is the same as the Tractor holonomy of the underlying mani…
Characterizes conformal Killing tensors and their Killing scales.
problem Characterizing conformal Killing tensors and their Killing scales.
method Differential prolongation using conformally invariant tractor calculus.
result Provides an invariant characterisation of Einstein Killing scales.
We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach.
For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as aff…
We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Car…
A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geomet…
This paper studies lightlike Cartan geometries and their properties.
problem Understanding geometric structures on lightlike cones in spacetime.
method Develops Cartan geometries on the future lightlike cone of Lorentz-Minkowski spacetime.
result Lightlike Cartan geometries induce a lightlike metric and compatible structures.
Introduces tractors for basic examples and modern differential calculus.
problem None explicitly stated, focuses on introduction.
method Classical examples and modern invariant differential calculus.
result Introduction to tractors and related modern differential calculus.
Study of tractor bundles and spacelike immersions in Lorentzian manifolds.
problem Characterizing and understanding conformal tractor bundles and spacelike immersions.
method Extrinsic viewpoint, relating tractor bundles to spacelike immersions, reformulating equations in terms of spacelike immersion geometry.
result Every Riemannian conformal structure can be realized as a pullback of the tangent bundle of a Lorentzian ambient space.
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
Tractor Calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the (d+2)-di…
We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic probl…
A regular normal parabolic geometry of type G/P on a manifold M gives rise to sequences Di of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\na^\om$ on the corresponding tractor bundle V, where $\om$ is…
Defines a new conformally invariant Yang-Mills type energy for 6-manifolds.
problem No specific problem stated; focuses on defining a new energy.
method Defines a conformally invariant action S on gauge connections on a 6-manifold M, leading to higher-order conformally invariant Yang-Mills equations.
result The Euler-Lagrange equations of S provide a conformally invariant analogue of Yang-Mills equations, with special cases recovering known invariants.
We generalize the notion of planar bicycle tracks -- a.k.a. one-trailer systems -- to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the domains that are swept out by any given tractor/tractrix system. These …
We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector …
Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.
problem Projective compactness for torsion-free linear connections on a manifold.
method Introduce and study a weakening of projective compactness for torsion-free linear connections on a manifold.
result Induces projective structure on the boundary and relates to asymptotic forms in GR.
This paper is a study of the relationship between two constructions associated with Cartan geometries, both of which involve Lie algebroids: the Cartan algebroid, due to [Blaom A.D., Trans. Amer. Math. Soc. 358 (2006), 3651-3671], and tractor calculus [Cap A., Gover A.R., Trans. Amer. Math. Soc. 354 (2001), 1511-1548].
Proposes a new variational principle for Einstein gravity.
problem Formulating Einstein gravity as a gauge theory for the conformal group.
method First order formulation of conformal tractor geometry, variational principle based on abstract principal bundle.
result Provides first order field equations without requiring supplementary constraints.
On a conformal manifold, it is well known that parallel sections of the standard tractor bundle with non-vanishing scale are in 1-1 correspondence with solutions of the conformal Einstein equation. In 2 dimensions conformal geometry carries no local information but one can remedy this by equipping the surface with a Mö…
The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as p…
The article constructs differential operators for parabolic geometries.
problem Developing a machinery for differential operators in parabolic geometries.
method Starting from a relative tractor bundle, constructs a sequence of differential operators.
result Provides a resolution of a sheaf for many geometries.
The Killing tensor equation is a first order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any manifold equipped with an affine connection, and in this setting derive its prolo…
We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classic…
We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non--degenerate CR manifold of hypersurface type. In particular we construct and treat the basic ob…
The aim of this paper is to describe the geometry of conformal structures in Lorentzian signature, which admit a lightlike conformal Killing vector field whose corresponding adjoint tractor acts as complex structure on the standard tractor bundle of conformal geometry. Key to the treatment of this problem is CR-geometr…
BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor conn…
This paper studies the relation between two notions of holonomy on a conformal manifold. The first is the conformal holonomy, defined to be the holonomy of the normal tractor connection. The second is the holonomy of the Fefferman-Graham ambient metric of the conformal manifold. It is shown that the infinitesimal confo…
This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in [5] and then compute the prolongation …
If the conformal holonomy group Hol(T) of a simply connected space with conformal structure of signature (2p−1,2q−1) is reduced to $\U(p,q)$ then the conformal holonomy is already contained in the special unitary group $\SU(p,q)$. We present two different proofs of this statement, one using conformal trac…