Given a forward ( = usual) stochastic differential equation (SDE), we consider, in this paper, an associated backward SDE. Let E;s,t(x),t∈[s, ∞) be the solution of an SDE on a manifold M:with the initial condition ξs,s(x) =x. HereX0,…,Xrare smooth vector fields, (Bt1,…,Bt1) is a standard r-dimensional Brownian motion and o denotes the Stratonovich integral. We show that the solution E;s,tsatisfies the backward SDE:where ξstthe differential of the map Es,t(·)M→Mand [dcirc]Bsjdenotes the backward stochastic integral. The result is applied to getting a necessary and sufficient condition that the map ξs,t: defines a diffeomorphism ofMa.s.