Solutions of the sDiff(2)Toda equation with SU(2) Symmetry

We present the general solution to the Plebanski equation for an H-space that admits Killing vectors for an entire SU(2) of symmetries, which is therefore also the general solution of the sDiff(2)Toda equation that allows these symmetries. Desiring these solutions as a bridge toward the future for yet more general solutions of the sDiff(2)Toda equation, we generalize the earlier work of Olivier, on the Atiyah-Hitchin metric, and re-formulate work of Babich and Korotkin, and Tod, on the Bianchi IX approach to a metric with an SU(2) of symmetries. We also give careful delineations of the conformal transformations required to ensure that a metric of Bianchi IX type has zero Ricci tensor, so that it is a self-dual, vacuum solution of the complex-valued version of Einstein's equations, as appropriate for the original Plebanski equation.Comment: 27 page

Classification of the Traceless Ricci Tensor in 4-dimensional Pseudo-Riemannian Spaces of Neutral Signature

The traceless Ricci tensor $C_{ab}$ in 4-dimensional pseudo-Riemannian spaces equipped with the metric of the neutral signature is analyzed. Its algebraic classification is given. This classification uses the properties of $C_{ab}$ treated as a matrix. The Petrov-Penrose types of Pleba\'nski spinors associated with the traceless Ricci tensor are given. Finally, the classification is compared with a similar classification in the complex case

Complex windmill transformation producing new purely magnetic fluids

Minimal complex windmill transformations of G2IB(ii) spacetimes (admitting a two-dimensional Abelian group of motions of the so-called Wainwright B(ii) class) are defined and the compatibility with a purely magnetic Weyl tensor is investigated. It is shown that the transformed spacetimes cannot be perfect fluids or purely magnetic Einstein spaces. We then determine which purely magnetic perfect fluids (PMpfs) can be windmill-transformed into purely magnetic anisotropic fluids (PMafs). Assuming separation of variables, complete integration produces two, algebraically general, G2I-B(ii) PMpfs: a solution with zero 4-acceleration vector and spatial energy–density gradient, previously found by the authors, and a new solution in terms of Kummer's functions, where these vectors are aligned and non-zero. The associated windmill PMafs are rotating but non-expanding. Finally, an attempt to relate the spacetimes to each other by a simple procedure leads to a G2I-B(ii) one-parameter PMaf generalization of the previously found metric