  New rigidity results for complete non-compact spacelike submanifolds of
arbitrary codimension in plane fronted waves are obtained. Under appropriate
assumptions, we prove that a complete spacelike submanifold in these spacetimes
is contained in a characteristic lightlike hypersurface. Moreover, for a
complete codimension two extremal submanifold in a plane fronted wave we show
sufficient conditions to guarantee that it is a (totally geodesic) wavefront.
