Vacuum Spacetimes of Embedding Class Two

Doubt is cast on the much quoted results of Yakupov that the torsion vector in embedding class two vacuum space-times is necessarily a gradient vector and that class 2 vacua of Petrov type III do not exist. The first result is equivalent to the fact that the two second fundamental forms associated with the embedding necessarily commute and has been assumed in most later investigations of class 2 vacuum space-times. Yakupov stated the result without proof, but hinted that it followed purely algebraically from his identity: R(ijkl)C(kl) = 0 where C(ij) is the commutator of the two second fundamental forms of the embedding. From Yakupov's identity, it is shown that the only class two vacua with non-zero commutator C(ij) must necessarily be of Petrov type III or N. Several examples are presented of non-commuting second fundamental forms that satisfy Yakupovs identity and the vacuum condition following from the Gauss equation; both Petrov type N and type III examples occur. Thus it appears unlikely that his results could follow purely algebraically. The results obtained so far do not constitute definite counter-examples to Yakupov's results as the non-commuting examples could turn out to be incompatible with the Codazzi and Ricci embedding equations. This question is currently being investigated.