On characterization of Dirichlet-to-Neumann map of Riemannian surface with boundary

Let $(M,g)$ be a smooth compact orientable two-dimensional Riemannian manifold ({\it surface}) with a smooth metric tensor $g$ and smooth connected boundary $Γ$. Its {\it DN-map} $Λ_g:{C^\infty}(Γ)\to{C^\infty}(Γ)$ is associated with the (forward) elliptic problem $Δ_gu=0 \,\,\, {\rm in}\,\,M\setminusΓ,\,\,u=f \,\,\, {\rm on}\,\,\,Γ$, and acts by $Λ_g f:=\partial_νu^f \,\,\, {\rm on}\,\,\,Γ,$ where $Δ_g$ is the Beltrami-Laplace operator, $u=u^f(x)$ is the solution, $ν$ is the outward normal to $Γ$. The corresponding {\it inverse problem} is to determine the surface $(M,g)$ from its DN-map $Λ_g$. We provide the necessary and sufficient conditions on an operator acting in ${C^\infty}(Γ)$ 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.

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