Rotation of a scroll wave reveals the metric tensor for the associated geodesic filament

This study considers a reaction-diffusion medium that supports a rotating scroll wave with stationary filament and diffusivity matrix D. The filament is a geodesic curve in three-space. The metric tensor g associated with that geodesic is obtained in terms of D, resulting in g=(detD)D-1. Here detD (the determinant of D) is not necessarily constant in space. The detD factor, although strongly indicated previously, has not yet been derived deductively, as it is here. We take the opportunity to outline, starting from the reaction-diffusion equations, a sequence of mathematical steps leading to both geodesic and metric.