On the EIT problem for nonorientable surfaces

Abstract

Let $(Ω,g)$ be a smooth compact two-dimensional Riemannian manifold with boundary, $Λ_g: f\mapsto \partial_νu|_{\partialΩ}$ its DN map, where $u$ obeys $Δ_g u=0$ in $Ω$ and $u|_{\partial Ω}=f$. The Electric Impedance Tomography problem is to determine $Ω$ from $Λ_g$. A criterion is proposed that enables one to detect (via $Λ_g$) 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 $Λ_g$ up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g')$ of $(M,g)$. This copy is conformally equivalent to the original, provides $\partial M'=\partial M,\,\,Λ_{g'}=Λ_g$, and thus solves the problem.