The validity of Huygens' principle for the conformally invariant scalar wave equation, Maxwell's equations and Weyl's neutrino equation on Petrov type D and type III space-times

It is shown that there exist no Petrov type D space-times on which the conformally invariant scalar wave equation, or Maxwell's equations or Weyl's neutrino equation satisfies Huygens' principle. For Petrov type III space-times, it is shown that the validity of Huygens' principle for any of these equations implies that the space-time is conformally related to one in which the repeated principal null vector field of the Weyl tensor is recurrent.