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 connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on (k − ℓ)-forms for various integers ℓ. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.

On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian ∆ k , k ∈ Z > 0, on Euclidean space E n . The algebra formed by the symmetry operators is described explicitly.

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 ∇ ω on the corresponding tractor bundle V, where ω is the normal Cartan connection. The first operator D0 in the sequence is overdetermined and it is well known that ∇ ω yields the prolongation of this operator in the homogeneous case M = G/P . Our first main result is the curved version of such a prolongation. This requires a new normalization∇ of the tractor covariant derivative on V . Moreover, we obtain an analogue for higher operators Di. In that case one needs to modify the exterior covariant derivative d ∇ ω by differential terms. Finally we demonstrate these results on simple examples in projective and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.

No abstract

No abstract