The notion of $F$-geodesic, which is slightly different from that of $F$-planar curve (see [13], [17], and [18]), generalizes the magnetic curves, and implicitly the geodesics, by using any (1,1)-tensor field on the manifold (in particular the electro-magnetic field or the Lorentz force). We give several examples of $F$-geodesics and the characterizations of the $F$-geodesics w.r.t. Vranceanu connections on foliated manifolds and adapted connections on almost contact manifolds. We generalize the classical projective transformation, holomorphic-projective transformation and $C$-projective transformation, by considering a pair of symmetric connections which have the same $F$-geodesics. We deal with the transformations between such two connections, namely $F$-planar diffeomorphisms ([18]). We obtain a Weyl type tensor field, invariant under any $F$-planar diffeomorphism, on a 1-codimensional foliation.