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

Fix xi a unitary vector field on a Riemannian manifold M and gamma a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that gamma is a Legendre curve for xi if and only if the gamma-Fermi-Walker covariant derivative of xi vanishes. The cases when gamma is circle or helix as well as xi 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.