Conformal representation of Kerr space–time poloidal sub-manifolds

We use two conformal transformations to represent field lines in the poloidal sub-manifold of Kerr space–time. The first one is based on an embedding in $R^3$ of a manifold which is conform to the poloidal submanifold. The second one is a planar representation using quasi-isotropic coordinates. We compare plots of the poloidal magnetic field lines in the usual Boyer–Lindquist Cartesian coordinates and in the conformal representation based on quasi-isotropic coordinates. In a conformal representation these lines appear entering in the horizon perpendicularly to it as deduced through a mathematical perspective conversely to the usual physical approach. We also compare the value of the conformal factor in these two representations. (paper)