The $P^\prime$-operator, the $Q^\prime$-curvature, and the CR tractor calculus

Conformal theory of curves with tractors

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 classical Frenet frame construction and then expressed in terms of relative ones. This approach applies likewise to conformal structures of any signature; in the case of indefinite signature we focus especially on the null curves. It also provides a conceptual tool for handling distingu…

The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry

We elaborate the tractor calculus for compatible almost CR structures (also known as strictly pseudoconvex partially integrable almost CR structures) on contact manifolds, and as an application, express the first BGG invariant differential operator $D_0$ explicitly in some cases, i.e., for some tractor connections. An interesting outcome is the fact that the "modified" adjoint tractor connection $\tilde{\nabla}$ governing infinitesimal deformations of parabolic geometries generates what we call the CR Killing operator as its first BGG operator, and actually, it does not agree with the first BG…

CR embedded submanifolds of CR manifolds

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 Cartan (or tractor) connections via a CR Gauss formula. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation o…

Tractor calculus, BGG complexes, and the cohomology of Kleinian groups

For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M= Γ\backslash \mathbb{H}^{n}$ where $Γ$ is a torsion-free cocompact subgroup of $SO(n,1)$, we establish and study a relationship between differential geometric cohomology on $M$ and algebraic invariants of the group $Γ$. In particular for $\mathbb{F}$ an irreducible $SO(n,1)$-module, we show that the group cohomology with coefficients $H^{\bullet}(Γ, \mathbb{F})$ is isomorphic to the cohomology of an appropriate projective BGG complex on $M$. This yields the geometric interpretation that $H^{\bullet}(Γ, \mathbb{F})$ parame…

Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk

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 problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter al…

Tractors, Mass and Weyl Invariance

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invarian…

射影无穷与b-微积分

Projective Infinities and b-Calculus

本文提出射影预紧性概念，证明在弱曲率条件下边界诱导射影结构，并利用b-切丛和密度丛的扩展得到标准拖拉机丛，进而将伪黎曼度量与广义相对论中的渐近形式联系起来。

An ambient approach to conformal geodesics

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article we give a third description using ideas from holography. A conformal $n$-manifold $X$ can be seen (formally at least) as the asymptotic boundary of a Poincaré--Einstein $(n+1)$-manifold $M$. We show that any curve $γ$ in $X$ has a uniquely determined extension to a surface $Σ_γ$ …

Sharp metric obstructions for quasi-Einstein metrics

Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor $W$ to the setting of smooth metric measure spaces. The obstructions we obtain can be realized as tensorial invariants which are polynomial in the Riemann curvature tensor and its divergence. By taking suitable limits of their tensorial forms, we then find obstructions to the existence of static potentials, generalizing to high…

Asymptotic Shear and the Intrinsic Conformal Geometry of Null-Infinity

In this article we propose a new geometrization of the radiative phase space of asymptotically flat space-times: we show that the geometry induced on null-infinity by the presence of gravitational waves can be understood to be a generalisation of the tractor calculus of conformal manifolds adapted to the case of degenerate conformal metrics. It follows that the whole formalism is, by construction, manifestly conformally invariant. We first show that a choice of asymptotic shear amounts to a choice of linear differential operator of order two on the bundle of scales of null-infinity. We refer t…

CR Invariant powers of the sub-Laplacian

CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ``conformally invariant powers of the Laplacian'' via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. A second family is derived from a CR tractor calculus which is developed here; this family includes operators for every positive power of the sub-Laplacian. This result together with work of Cap, Slovak and Soucek imply in three dimensions the existence of a curved analogue of ea…

The almost Einstein operator for $(2, 3, 5)$ distributions

For the geometry of oriented $(2, 3, 5)$ distributions $(M, {\mathbf D})$, which correspond to regular, normal parabolic geometries of type $(\mathrm{G}_2, P)$ for a particular parabolic subgroup $P < \mathrm{G}_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator $Θ_0$ associated to the $7$-dimensional irreducible representation of $\mathrm{G}_2$. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator are automat…