Notes on the Geometry of Electromagnetic Fields and Maxwell's Equations along a non-null curves in non flat-3D space forms $M_{q}^{3}(c)$
Fatma Almaz, Cumali Ekici
TL;DR
The paper addresses electromagnetic fields and Maxwell's equations along non-null curves in non-flat 3D space forms $M_{q}^{3}(c)$ by developing a Frenet-Serret framework in curved spaces with frame $\{T,N,B\}$ and a gradient operator $\nabla = T\partial_{s}+N\partial_{\xi}+B\partial_{\eta}$. It derives extended Serret-Frenet relations, computes Div and Curl of the frame fields, and explores compatibility conditions that tie curvature $\kappa$ and torsion $\tau$ to frame coefficients within the Sasaki metric on $T M_{q}^{3}(c)$. Maxwell's equations are formulated for the electric and magnetic fields along fiber-like curves, with Berry-phase effects and polarization rotations described in terms of the $\{T,N,B\}$ frame and Lorentz-force-type relations. The bending-energy functionals for vector fields are then obtained for the $s$-, $\xi$-, and $\eta$-directions, yielding explicit expressions in terms of geometric invariants ($\kappa,\tau$) and differential operators ($\mathrm{Curl},\mathrm{Div}$) on $N$ and $B$, providing a geometric electromagnetism framework in curved 3-manifolds applicable to optical-fiber-like settings in non-Euclidean spaces.
Abstract
In this paper, the directional derivatives in accordance with the orthonormal frame {T, N, B} are defined in $M_{q}^{3}(c)$, and the extended Serret-Frenet relations by using Frenet formulas are expressed. Furthermore, we express the bending elastic energy function for the same particle in $M_{q}^{3}(c)$ according to curve $α(s,ξ,η)$ and geometrical interpretation of the energy for unit vector fields and we also solve Maxwell's equations for the electric and magnetic field vectors in $M_{q}^{3}(c).$