On the EIT problem for nonorientable surfaces

Let {(\Omega,g)} be a smooth compact two-dimensional Riemannian manifold with boundary and let {\Lambda_{g}:f\mapsto\partial_{\nu}u|_{\partial\Omega}} be its DN map, where u obeys {\Delta_{g}u=0} in Ω and {u|_{\partial\Omega}=f}. The Electric Impedance Tomography Problem is to determine Ω from {\Lambda_{g}}. A criterion is proposed that enables one to detect (via {\Lambda_{g}}) whether Ω is orientable or not. The algebraic version of the BC-method is applied to solve the EIT problem for the Moebius band. The main instrument is the algebra of holomorphic functions on the double covering {{\mathbb{M}}} of M, which is determined by {\Lambda_{g}} up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy {(M^{\prime},g^{\prime})} of {(M,g)}. This copy is conformally equivalent to the original, provides {\partial M^{\prime}=\partial M}, {\Lambda_{g^{\prime}}=\Lambda_{g}}, and thus solves the problem.