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Self-Dual Conformal Gravity | | Maciej Dunajski
; Paul Tod
; | | Date: |
29 Apr 2013 | | 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 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 $eta$-surfaces. | | Source: | arXiv, 1304.7772 | | Services: | Forum | Review | PDF | Favorites | |
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