  In pure Einstein theory, Ricci flat Lorentzian 4-metrics of Petrov types III
or N have vanishing counter terms up to and including two loops. Moreover for
pp-waves and type-N spacetimes of Kundt's class which admit a non-twisting, non
expanding, null congruence all possible invariants formed from the Weyl tensor
and its covariant derivatives vanish. Thus these Lorentzian metrics suffer no
quantum corrections to all loop orders. By contrast for complete non-singular
Riemannian metrics the two loop counter term vanishes only if the metric is
flat.
