USvia a ber respecting dieomorphism:EjUw[[[]pUSpr1UEis called thetotal space,Mis called thebase space,pis a surjective submersion,called theprojection, andSis calledstandard ber. (U; ) as above is called aber chartor alocal trivializationofE.

A(ber) bundle(E;p;M;S) consists of manifoldsE,M,S,and a smooth mappingp:E!M; furthermore it is required that eachx2Mhas an open neighborhoodUsuch thatEjU:=p1(U) is dieomorphic to