In this article, we introduce a process to reconstruct a Riemann surface with boundary equipped with a linked conductivity tensor from its boundary and the Dirichlet-Neumann operator associated to this conductivity. When initial data comes from a two dimensional real Riemannian surface equipped with a conductivity tensor, this process recovers its conductivity structure. Dedication In January 2016 my friend Gennadi Henkin, with whom I had worked for more than …fteen years, passed away. This paper, which comes back on a subject he brought, is dedicated to him. The numerous citations from the articles he authored show the depth of the mathematical thought of Gennadi.