Communications in Analysis and Geometry
Volume 20 (2012)
Scattering boundary rigidity in the presence of a magnetic field
Pages: 501 – 528
It has been shown in  that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric results in  to regions with the same scattering data, where only one is known a priori to be simple. We will then use this results to resolve a local version of a question by Robert Bryant. That is, we show that a surface of constant curvature cannot be modified in a small region while keeping all the curves of some fixed constant geodesic curvatures closed.