Abstract: BGG-operators form sequences of natural differential operators on bundles over parabolic geometries. Many interesting geometric equations are defined by the first operators in such sequences: for instance, in conformal geometry, the spaces of Einstein scales, conformal Killing fields and conformal Killing forms appear as kernels of certain BGG-operators.

We discuss natural geometric prolongations of such BGG-equations: i.e., we construct a natural linear connection whose space of parallel sections is canonically isomorphic to the kernel of the first BGG-operator. As a consequence, we obtain natural tensorial objects which obstruct the existence of non-trivial solutions.