Related papers: The CR Killing operator and Bernstein-Gelfand-Gelf…
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…
For a manifold $M$ endowed with a Legendrean (or Lagrangean) contact structure $E\oplus F \subset TM$, we give an elementary construction of an invariant partial connection on the quotient bundle $TM/F$. This permits us to develop a…
We study the standard tractor bundle and the standard cotractor bundle of an almost Grassmann structure: We provide explicit formulae for their splitting operators, first BGG operators as well as prolongation connections. We characterize…
We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR…
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…
We address the problem of how to characterise when a rank-two conformal Killing tensor is the trace-free part of a Killing tensor for a metric in the conformal class. We call such a metric a Killing scale. Our approach is via differential…
We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing--Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to…
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…
For a real or complex semisimple Lie group $G$ and two nested parabolic subgroups $Q\subset P\subset G$, we study parabolic geometries of type $(G,Q)$. Associated to the group $P$, we introduce a class of relative natural bundles and…
In linearized gravity, two linearized metrics are considered gauge-equivalent, $h_{ab} \sim h_{ab} + K_{ab}[v]$, when they differ by the image of the Killing operator, $K_{ab}[v] = \nabla_a v_b + \nabla_b v_a$. A universal (or complete)…
A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal…
This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated to the geometry in…
We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is a type of moving incidence relation. The characterisation is used to provide…
When ${\cal{D}}: E \rightarrow F$ is a linear differential operator of order $q$ between the sections of vector bundles over a manifold $X$ of dimension $n$, it is defined by a bundle map $\Phi: J_q(E) \rightarrow F=F_0$ that may depend,…
We provide an extension of the Gromov--Zimmer Embedding Theorem for Cartan geometries of [3] to tractor bundles carrying any invariant connection, including tractor connections and prolongation connections of first BGG operators for…
This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features…
We establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both…
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…
BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability…
The purpose of this paper is to describe certain CR-covariant differential operators on a strictly pseudoconvex CR manifold $M$ as residues of the scattering operator for the Laplacian on an ambient complex K\"{a}hler manifold $X$ having…