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Let(  Ω  ,  g  ){(\Omega,g)}    be a smooth compact two-dimensional Riemannian manifold with boundary and letΛ  g   :   f  ↦∂  ν   ⁡  u   |    ∂  ⁡  Ω{\Lambda_{g}:f\mapsto\partial_{\nu}u|_{\partial\Omega}}    be its DN map, where  u  obeysΔ  g   ⁢  u   =  0{\Delta_{g}u=0}    in Ω andu  |    ∂  ⁡  Ω    =  f{u|_{\partial\Omega}=f}   . The Electric Impedance Tomography Problem is to determine Ω fromΛ  g{\Lambda_{g}}   . A criterion is proposed that enables one to detect (viaΛ  g{\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 byΛ  g{\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  ′   ,   g  ′   ){(M^{\prime},g^{\prime})}    of(  M  ,  g  ){(M,g)}   . This copy is conformally equivalent to the original, provides∂  ⁡   M  ′    =   ∂  ⁡  M{\partial M^{\prime}=\partial M}   ,Λ   g  ′    =   Λ  g{\Lambda_{g^{\prime}}=\Lambda_{g}}   , and thus solves the problem.

On the EIT problem for nonorientable surfaces