On the EIT problem for nonorientable surfaces

TL;DR

This paper develops a criterion to detect whether a surface is orientable using the DN map in EIT and applies algebraic methods to solve the inverse problem for the Möbius band, advancing understanding of nonorientable surface reconstruction.

Contribution

It introduces a criterion for orientability detection via the DN map and applies the algebraic BC-method to solve the EIT problem on the Möbius band, a nonorientable surface.

Findings

Criterion for detecting nonorientability via DN map

Solution of EIT for the Möbius band

Construction of conformal copies from algebraic data

Abstract

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.