called quantomorphisms, which cover the 1-parameter group e tX f . 2 In applications, we have to deal with incomplete vector fields. In this case, e tX f and e tZ f are local 1-parameter groups of local diffeomorphisms of P and L × , respectively. In order to avoid this lengthy expression, we shall refer to e tX f and e tZ f as flows of X f and Z f , respectively. though it is an abuse of terminology. Since L is an associated bundle of L × , the local action e tZ f : L × → L × , induces a local action e tZ f : L → L, which acts on smooth sections σ of L by push forwards, σ → e. Though e tZ f * σ may not be defined for all σ and all t, its derivative at t = 0 is defined for all smooth sections.R f σ = d dt t=0 e tZ f * σ.