Mikołaj Korzyński and Jerzy Lewandowski 2003 Class. Quantum Grav. 20 3745 doi:10.1088/0264-9381/20/16/314
Mikołaj Korzyński1 and Jerzy Lewandowski1,2,3
Show affiliationsThe goal of this paper is to express the Bach tensor of a four-dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang–Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach tensor. As an example we study the case of a CNC connection which admits a twisting covariantly constant twistor field. This class of conformal geometries of this property is known as given by the Fefferman metric tensors. We use our result to calculate the Bach tensor of an arbitrary Fefferman metric and show that it is proportional to the tensorial square of the four-fold eigenvector of the Weyl tensor. Finally, we solve the Yang–Mills equations imposed on the CNC connection for all the homogeneous Fefferman metrics. The only solution is the Nurowski–Plebański metric.
Issue 16 (21 August 2003)
Received 11 February 2003
Published 31 July 2003
Mikołaj Korzyński and Jerzy Lewandowski 2003 Class. Quantum Grav. 20 3745
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