Abstract:
We develop a universal and algorithmic construction of invariant differential operators between irreducible bundles in conformal geometry. The classification of such operators in the flat case is well-known in terms of representation theory. The main result of the thesis is a construction of curved analogues of these. We obtain curved analogues in every case save for an exception which exists in every pattern in every even dimension. The operators are described via explicit formulae in tractor calculus. These are closely related to the usual “V-formulae” for invariant operators in Riemannian geometry. The construction follows Eastwood’s curved translation principle which we implement in the conformal tractor calculus. We work in both real and complex setting and for all signatures. Further, we use the developed calculus to study one class of these operators - the conformal Killing operator on forms - in detail. We construct invariant prolongations of the corresponding systems of partial differential equations. Using these, we obtain information about the solution space. In particular, we develop a helicity raising and lowering construction in the general setting, and also on conformally Einstein manifolds.