  Let $(M,g)$ be a smooth compact orientable two-dimensional Riemannian
manifold ({\it surface}) with a smooth metric tensor $g$ and smooth connected
boundary $\Gamma$. Its {\it DN-map}
$\Lambda_g:{C^\infty}(\Gamma)\to{C^\infty}(\Gamma)$ is associated with the
(forward) elliptic problem $ \Delta_gu=0 \,\,\, {\rm
in}\,\,M\setminus\Gamma,\,\,u=f \,\,\, {\rm on}\,\,\,\Gamma$, and acts by
  $ \Lambda_g f:=\partial_\nu u^f \,\,\, {\rm on}\,\,\,\Gamma,
  $ where $\Delta_g$ is the Beltrami-Laplace operator, $u=u^f(x)$ is the
solution, $\nu$ is the outward normal to $\Gamma$. The corresponding {\it
inverse problem} is to determine the surface $(M,g)$ from its DN-map
$\Lambda_g$.
  We provide the necessary and sufficient conditions on an operator acting in
${C^\infty}(\Gamma)$ to be the DN-map of a surface. In contrast to the known
conditions by G.Henkin and V.Michel in terms of multidimensional complex
analysis, our ones are based on the connections of the inverse problem with
commutative Banach algebras.
