Harmonic mappings of Riemannian manifolds and stationary vacuum space–times with whole cylinder symmetry

We consider stationary, cylindrically‐symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton–Jacobi techniques, we obtain the geodesics and construct the resulting space–time geometries. We find that the light cone structure of the configuration space delineates the distinct exterior fields of Lewis and van Stockum which together form the most general solution with whole cylinder symmetry.