Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Self-Dual Conformal Gravity   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.

29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn