  The formulation of the Einstein field equations admitting two Killing vectors
in terms of harmonic mappings of Riemannian manifolds is a subject in which
Charlie Misner has played a pioneering role. We shall consider the hyperbolic
case of the Einstein-Maxwell equations admitting two hypersurface orthogonal
Killing vectors which physically describes the interaction of two electrovac
plane waves. Following Penrose's discussion of the Cauchy problem we shall
present the initial data appropriate to this collision problem. We shall also
present three different ways in which the Einstein-Maxwell equations for
colliding plane wave spacetimes can be recognized as a harmonic map. The goal
is to cast the Einstein-Maxwell equations into a form adopted to the initial
data for colliding impulsive gravitational and electromagnetic shock waves in
such a way that a simple harmonic map will directly yield the metric and the
Maxwell potential 1-form of physical interest.
  For Charles W. Misner on his 60th birthday

General Relativity and Quantum Cosmology