On the existence of the field line solutions of the Einstein-Maxwell equations

The main result of this paper is the proof that there are local electric and magnetic field configurations expressed in terms of field lines on an arbitrary hyperbolic manifold. This electromagnetic field is described by (dual) solutions of the Maxwell's equations of the Einstein-Maxwell theory. These solutions have the following important properties: i) they are general, in the sense that the knot solutions are particular cases of them and ii) they reduce to the electromagnetic fields in the field line representation in the flat space-time. Also, we discuss briefly the real representation of these electromagnetic configurations and write down the corresponding Einstein equations.