Mh. Math. 105, 287--293 (1988) Iv~~ffir Malhemalik 9 by Springer-Verlag 1988 A Lie Group Structure on the Space of Time-dependent Vector Fields By Andrea Posilicano, Milano (Received 29 May 1987) Abstract. In this note we use a banal relation between flows of vector fields, a sort of time-dependent Campbell--Baker--Hausdorff formula, to obtain a non-linear version of the variation of constants formula. We employ this formula to calculate the tangent to the mapping which assigns to each time-dependent vector field V with compact support the diffeomorphism q~, (V), where ~ (V) is the global flow of V. Afterwards we use these results to give a non-commutative Lie group structure to the space of time-dependent vector fields with compact support. We also treat of the Lie algebra of this group and we calculate the adjoint action and the exponential mapping. The setting is the same of [1] and we refer the reader to this book for notations. 1. Let V be a time-dependent vector field and let ~bt,~(V) be the evolution operator generated by V according to d q~t,~(V)(m) = V(t,~b,,~(V)(m)) q~,~(V)(rn)= m. By uniqueness of integral curves we have qSt,s(g) o~s,r(g ) = ~bt, r(g ) qDs, s ( g ) ~-- id. We will write ~bt(V) = ~bt, o(V) and q~(v) = q~l(V). We observe that we have ~,,s(V) = ~,(V)o~s(V) -1 If the vector field X is time-independent we have ~bt, s(X ) = q~t-s(X) and we will write exp t X = ~b,(X). Taking the time derivative of exp t Xo exp t Y and using the definition of flow we obtain exp t Xo exp t Y = 05 t (X. I1)