2010/01/11 by Daniel Finley, John K. McIver, J. K. McIver
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Conformal map #Geometric Analysis and Curvature Flows #Geometry #Homogeneous space #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Nonlinear Waves and Solitons #Physics #Pure mathematics #Space (punctuation) #Symmetry (geometry) #Tensor (intrinsic definition) #gr-qc
paper · pdf · doi:10.1088/0264-9381/27/14/145001
published as Class.Quant.Grav.27:145001,2010 · 27 pages
arxiv created 2010/01/11 · openalex publication_date 2010/05/26 · arxiv updated 2014/11/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.