The Birkhoff Theorem in Multidimensional Gravity
  The validity conditions for the extended Birkhoff theorem in multidimensional gravity with $n$ internal spaces are formulated, with no restriction on space-time dimensionality and signature. Examples of matter sources and geometries for which the theorem is valid are given. Further generalization of the theorem is discussed. 
K.A.Bronnikov and V.N.Melnikov