Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

Abstract

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

Normal Fermi- Walker Derivative

“…Also, unit vector fields according to Fermi-Walker transported are proved along Rytov-Legendre curves. Slant and Legendre curves are defined in three-dimensional warped products and are characterised via the scalar products between the normal of these curves and the vertical vector field [5], [4]. The constant precession curve is expressed as unit-speed curve that its central returns approximate a stable axis with stationary angle and stationary speed.…”

The flow-curvature of spacelike parametrized curves in the Lorentz plane

“…An important tool in dynamics is the Fermi-Walker derivative. Then the Lorentz Fermi-Walker derivative is the hyperbolic variant of derivative from[6], namely ∇…”