On the EIT problem for nonorientable surfaces

Abstract

Abstract Let ( Ω , g ) {(\Omega,g)} be a smooth compact two-dimensional Riemannian manifold with boundary and let Λ g : f ↦ ∂ ν ⁡ u | ∂ ⁡ Ω {\Lambda_{g}:f\mapsto\partial_{\nu}u|_{\partial\Omega}} be its DN map, where u obeys Δ g ⁢ u = 0 {\Delta_{g}u=0} in Ω and u | ∂ ⁡ Ω = f {u|_{\partial\Omega}=f} . The Electric Impedance Tomography Problem is to determine Ω from Λ g {\Lambda_{g}} . A criterion is proposed that enables one to detect (via Λ 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 𝕄 {{\mathbb{M}}} of M, which is determined by Λ 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.