)A principal gauge bundle "P" over a 4-manifold "M" is locally isomorphic to "U" × "F", where "U" ⊂ R4 and the fiber "F" is isomorphic to a Lie group "G", the gauge group of the field theory (this is an isomorphism of manifold structures, not of group structures; there is no special surface in "P" corresponding to 1 in "G", so it is more proper to say that the fiber "F" is a "G"-torsor).