Abstract : Any open Riemann surface has a conformal model in any orientable Riemannian manifold of dimension 4. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable 4-dimensional Riemannian manifold. This result along with \cite{ko6} now shows that an open Riemann surface admits conformal models in any Riemannian manifold of dimension $\ge 3.$
