2003/01/31 by Mikolaj Korzynski, Miko aj Korzy ski, Jerzy Lewandowski · 30 citations
Mathematics · Medicine · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Neuroimaging Techniques and Applications #Conformal field theory #Conformal geometry #Conformal map #Connection (principal bundle) #Exact solutions in general relativity #Geodesic #Geometric Analysis and Curvature Flows #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Metric tensor #Physics #Pure mathematics #Quantum mechanics #Riemann curvature tensor #Tensor (intrinsic definition) #Tensor field #Weyl tensor #Weyl transformation #gr-qc
paper · pdf · doi:10.1088/0264-9381/20/16/314
published in Classical and Quantum Gravity 20(16), 3745-3764 (IOP Publishing) · 30 pages, no figures, LaTeX, to be published in Class. Quant. Grav
arxiv created 2003/06/24 · openalex publication_date 2003/07/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The 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.