Related papers: Invariant Prolongation of BGG-Operators in Conform…
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…
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…
We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this…
This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the…
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 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…
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…
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…
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…
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…
We review the theory of first BGG operators and study how to approach them and find their solution on homogeneous geometries. We provide many new examples of parabolic geometries that admit solutions of first BGG operators with many…
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…
The aim of this work is to develop a systematic manner to close overdetermined systems arising from conformal Killing tensors (CKT). The research performs this action for 1-tensor and 2-tensors. This research makes it possible to develop a…
We consider generalized gradients in the general context of $G$-structures. They are natural first order differential operators acting on sections of vector bundles associated to irreducible $G$-representations. We study their geometric…
Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of…
The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed.…
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…
We discuss conformally covariant differential operators, which under local rescalings of the metric, \delta_\sigma g^{\mu\nu} = 2 \sigma g^{\mu\nu}, transform according to \delta_\sigma \Delta = r \Delta \sigma + (s-r) \sigma \Delta for…
We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting…
BGG resolutions and generalized BGG resolutions from representation theory of semisimple Lie algebras have been generalized to sequences of invariant differential operators on manifolds endowed with a geometric structure belonging to the…