Further applications of harmonic mappings of Riemannian manifolds to gravitational fields

We consider static, spherically-symmetric and stationary, axially-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. For static space--times with spherical symmetry we obtain the well-known solutions of Schwarzschild, Reissner--Nordstroem, and Janis--Newman--Winicour, where in the latter two cases the source for the geometry is the static, source-free Maxwell field and the massless scalar field respectively. When we consider the source to be a triad of massless scalar fields with the internal symmetry SU(2) x SU(2), we find that the space--time geometry is once again that of Janis--Newman--Winicour. Finally we formulate the stationary, axisymmetric gravitational field problem in terms of composite mappings and obtain the Weyl, Papapetrou, and the generalized Lewis and van Stockum classes of axisymmetric solutions