AbstractLet {(\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.