A Pasting Lemma I: the case of vector fields

We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that C volume preserving vector fields are C-dense in C volume preserving vector fields. Moreover, we obtain that C robustly transitive conservative flows in threedimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of the version of our pasting lemma for conservative diffeomorphisms, we show that C-robustly transitive conservative C-diffeomorphisms admits a dominated splitting, thus solving a question posed by Bonatti-DiazPujals. In particular, stably ergodic diffeomorphisms admits a dominated splitting.