Self-Dual Conformal Gravity

Understand

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold $(M, g)$ with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold.

• These conditions are expressed as the vanishing of scalar and tensor conformal invariants.
• The invariants obstruct the existence of parallel sections of a certain connection on a complex rank-four vector bundle over $M$.
• They provide a natural generalisation of the Bach tensor which vanishes identically for anti-self-dual conformal structures.