Self-Dual Conformal Gravity

Abstract

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. We use the obstructions to demonstrate that LeBrun's anti-self-dual metrics on connected sums of $\CP^2$s are not conformally Ricci-flat on any open set. We analyze both Riemannian and neutral signature metrics. In the latter case we find all anti-self-dual metrics with a parallel real spinor which are locally conformal to Einstein metrics with non-zero cosmological constant. These metrics admit a hyper-surface orthogonal null Killing vector and thus give rise to projective structures on the space of $\beta$-surfaces.

22 pages. Sections about local twistor transport, and LeBrun metrics on connected sums partially rewritten. To appear in Communications in Mathematical Physics

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• Einstein Gravity from Conformal Gravity
• Metrisability of three-dimensional projective structures
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• Nonconformally Ricci-flat instantons in Conformal Gravity with and without nonlinear matter fields
• On Walker and para-Hermite Einstein spaces