We determine all conformally Einstein and Bach-flat left-invariant Lorentzian metrics on almost Abelian Lie groups ℝ^3⋊ℝ . This is the final step in the classification of conformally Einstein and Bach-flat four-dimensional Lorentzian Lie groups, complementing our previous studies on semi-direct extensions of the Heisenberg group, the Euclidean and Poincaré Lie groups, and the non-solvable ones SU(2)×ℝ and SL(2,ℝ)×ℝ .
Using gauge theory, we classify SU(m+1) -equivariant holomorphic maps from the projective space CP^m , equipped with the Fubini–Study metric, into the Grassmann manifold Gr_p(C^p+m+1) . We show that the moduli space of such maps, modulo image equivalence, can be regarded as a subset of the gauge equivalence classes of invariant connections on homogeneous vector bundles over CP^m satisfying semi-positivity.
