Riemann Extension of the Anti-Mach Space time
  Riemann extension for the anti Mach metric is derived, the solution of geodesic equations for the extended space are given, some properties for the extended space was studied and compared with the basic space and the constructions of a translation surface for the anti Mach metric in four dimension is established. 
M. Abdel-Megied, Nassar H. Abdel-All and E. A. Hegazy
math.DG