The Geometry of classical change of signature
  The proposal of the possibility of change of signature in quantum cosmology has led to the study of this phenomenon in classical general relativity theory, where there has been some controversy about what is and is not possible. We here present a new analysis of such a change of signature, based on previous studies of the initial value problem in general relativity. We emphasize that there are various continuity suppositions one can make at a classical change of signature, and consider more general assumptions than made up to now. We confirm that in general such a change can take place even when the second fundamental form of the surface of change does not vanish. 
M.Carfora, G.F.R.Ellis