Let $(\Omega,g)$ be a smooth compact two-dimensional Riemannian manifold with boundary, $\Lambda_g: f\mapsto \partial_\nu u|_{\partial\Omega}$ its DN map, where $u$ obeys $\Delta_g u=0$ in $\Omega$ and $u|_{\partial \Omega}=f$. The Electric Impedance Tomography problem is to determine $\Omega$ from $\Lambda_g$. A criterion is proposed that enables one to detect (via $\Lambda_g$) whether $\Omega$ 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 $(M',g')$ of $(M,g)$. This copy is conformally equivalent to the original, provides $\partial M'=\partial M,\,\,\Lambda_{g'}=\Lambda_g$, and thus solves the problem.

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关于不可定向表面的 EIT 问题

令 $(\Omega,g)$ 是一个光滑紧凑的二维黎曼流形，其边界为 $\Lambda_g： f\mapsto \partial_\nu u|_{\partial\Omega}$ 为其 DN 映射，其中 $u$在 $\Omega$ 和 $u|_{\partial \Omega}=f$ 中遵守 $\Delta_g u=0$。Electric Impedance Tomography 问题是从 $\Lambda_g$ 确定 $\Omega$。提出了一种标准，使人们能够检测（通过 $\Lambda_g$）$\Omega$ 是否可定向。BC 方法的代数版本用于解决 Moebius 带的 EIT 问题。主要工具是在 $M$ 的双覆盖 ${\mathbb M}$ 上的全纯函数的代数，由 $\Lambda_g$ 决定直到等距同构。它的 Gelfand 谱（字符集）起到了构建 $(M,g)$ 的相关副本 $(M',g')$ 的材料作用。