Maciej Dunajski, Paul Tod
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 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 the Riemannian and the neutral signature metrics. In the latter case we find all anti-self-dual metrics with 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.
View original:
http://arxiv.org/abs/1304.7772
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