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In this work, we study inextensible flows of tangent bimagnetic particles in space. Moreover, we obtain the inextensible flows of electric field by Lorentz equation. Finally, we obtain new results for inextensible flows of tangent bimagnetic particles in space.

In this study, we obtain the special type of magnetic trajectories associated with a magnetic field ${{𝒱}}_b$ defined on a 3D Riemannian manifold. We investigate a new representation of binormal spherical indicatrices of magnetic curves. Thus, we study $B_b$-magnetic curves terms of inextensible flows. Furthermore, we give some new characterizations of curvatures in terms of some partial differential equations. Finally, we examine some geometrical and physical features of the moving charged particle corresponding to the $B_b$-magnetic curves. Namely, we compute uniformity of the $B_b$-magnetic curves.

In this work, we compute the time evolution equations (TEEs) of the type-1 Bishop frame of the curve. Also, we study the time evolution equations for type-1 Bishop curvatures (TEEBCs) as a system of partial differential equations (PDEs). Through this study, we give a necessary and sufficient condition for the normal and binormal Bishop velocities. Also, we construct new models of normal motions of inextensible curves in ${ℝ}^3$. These models are constructed for curves that move according to the type-1 Bishop frame. Also, we make a comparison study between surfaces obtained by the motions of inextensible curves according to the two frames, the type-1 Bishop frame and the Frenet frame.

In this paper, we investigate the flow of curve and its equiform geometry in 4-dimensional Galilean space. We obtain that the Frenet equations and curvatures of inextensible flows of curves and its equiformly invariant vector fields and intrinsic quantities are independent of time. We find that the motions of curves and its equiform geometry can be defined by the inviscid and stochastic Burgers’ equations in 4-dimensional Galilean space.