Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D semi-Riemannian manifold

We review the geometric evolution of a linearly polarized light wave coupling into an optical fiber and the rotation of the polarization plane in the three dimensional semi-Riemannian manifold. The optical fiber is assumed to be a one-dimensional object imbedded in a 3D semi-Riemannian manifold along the paper. Thus, in the 3D semi-Riemannian manifold, we demonstrate that the evolution of a linearly polarized light wave is associated with the Berry phase or more commonly known as the geometric phase. The ordinary condition for parallel transportation is defined by the Fermi-Walker parallelism law. We define other Fermi-Walker parallel transportation laws and connect them with the famous Rytov parallel transportation law for an electric field , which is considered as the direction of the state of the linearly polarized light wave in the optical fiber in the 3D semi-Riemannian manifold. Later, we define a special class of magnetic curves called by Bishop electromagnetic curves (-curves), which are generated by the electric field along the linearly polarized monochromatic light wave propagating in the optical fiber. In this way, not only we define a special class of linearly polarized point-particles corresponding to -curves of the electromagnetic field along with the optical fiber in the 3D semi-Riemannian manifold, but we also calculate, both numerically and analytically, the electromagnetic force, Poynting vector, energy-exchanges rate, optical angular and linear momentum, and optical magnetic-torque experienced by the linearly polarizable point-particles along with the optical fiber in the 3D semi-Riemannian manifold.