Coordinate-free Grad-Shafranov equation on a Riemannian manifold with Killing field

The MHD force-balance equation is formulated on a Riemannian manifold with the vector potential, the magnetic field, the current density and the pressure gradient as differential forms. In the presence of a non-vanishing Killing field (isometry) and the assumption that the physics is invariant under its flow, a coordinate-free Grad-Shafranov equation (GSE) is derived for the interior product of the vector potential with the Killing field. The symmetry reduces the MHD equilibrium problem to a two-dimensional elliptic boundary value one, where flux-surfaces are effectively iso-contours extruding along the Killing flow. The usual axisymmetric tokamak GSE and its helical extension are obtained as special cases from Euclidean three-space. Examples of GSE on non-flat Riemannian manifolds are explored.