On metric-connection compatibility and the signature change of   space-time
  We discuss and investigate the problem of existence of metric-compatible linear connections for a given space-time metric which is, generally, assumed to be semi-pseudo-Riemannian. We prove that under sufficiently general conditions such connections exist iff the rank and signature of the metric are constant. On this base we analyze possible changes of the space-time signature. 
Bozhidar Z. Iliev (Institute for Nuclear Research and Nuclear Energy,   Bulgarian Academy of Sciences, Sofia, Bulgaria)