Hidden Symmetries of PLEBAŃSKI'S Equations

Details Hidden Symmetries of PLEBAŃSKI'S Equations Hidden Symmetries of PLEBAŃSKI'S Equations

Dec 2002

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2002nmgm.meet..803d&link_type=abstract

"THE NINTH MARCEL GROSSMANN MEETING On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and R

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Let {M} be an oriented complex four-manifold (also called space-time) equipped with a holomorphic metric g. Plebański 7 has shown that the anti-self-dual vacuum (hyper-Kähler) equations Φ ABA'B' = 0 R = 0 CA'B'C'D' = 0, (1) locally imply the existence of a complex-valued function Θ and a coordinate system (w, z, x, y) such that the metric is given by g = 2{d}w{d}x {+} {2d}z{{d}}y {-} {2}Θxx {d}z2 - 2Θyy {d}w2 + 4Θxy {d}w{d}z (2) and Θ satisfies the second heavenly equation Θxw + Θyz + Θxx Θyy - Θxy 2 = 0 (3) (Here R is the Ricci scalar, ΦABA'B' is the trace-free part of the Ricci tensor, and CA'B'C'D' is the self-dual part of the Weyl tensor).
The first part of my talk was a review of some old and new results about conformal symmetries of this equation and its generalisations 6,1,8,2,4,5.
In the second part of the talk I have analysed equations (1) subject to an additional constraint...

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