Geometric phases for three cases of the electric field with new type Bishop frame in ℝ13

Abstract

In this paper, we study geometric phases for an electric field in Minkowski 3-space. In particular, we consider Bishop 2-type frames of a spacelike and a timelike curve in Minkowski 3-space and investigate three geometric phases for three cases of electric fields with respect to the Bishop 2-type frames along a spacelike and a timelike curve. Also, we compute spatial and time evolutions of the complex frame connected with electric fields for Bishop 2-type in terms of modified Hasimoto transformations. As a result, we give a modified Fermi–Walker parallel derivative and a Rytov parallel transportation of electric fields. Finally, we discuss Lorentz force of a magnetic field as a Killing vector field along a spacelike and timelike curve in Minkowski 3-space.