Maxwellian evolution of elliptical Darboux surfaces with abnormalities and electromagnetic trajectories

Abstract Classical formulations of Maxwell’s equations offer limited insight into the complex interplay between electromagnetic fields and the geometry of curved surfaces. This study introduces a fundamentally new framework that integrates Maxwell’s equations with differential geometry through an elliptically adapted Darboux frame, providing a dynamic and geometry-aware interpretation of electromagnetic wave behavior. We define electromagnetic (E–M) curves as trajectories of the field or polarization vector expressed in this frame, and we derive their Maxwellian evolution along anholonomic directions. Within this framework, we identify and characterize entirely new families of trajectories—Darboux magnetic curves, Darboux electromagnetic curves, and Rytov–Darboux curves—revealing that slant helices emerge as special cases. Unlike previous studies that simply apply geometric tools to field theory, it reconstructs electromagnetic evolution within a fully differential-geometric context, uncovering deep structural connections between field dynamics and surface curvature. This work positions itself at the frontier of mathematical physics, contributing a decisive step toward unifying field theory and differential geometry in the study of electromagnetic phenomena.