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.

We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. Studying the compatibility and the anti-compatibility relations between the determined structures and a natural diagonal metric, we find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. Finally, we prove the characterization theorem for the natural diagonal (almost) para-Kählerian structures on the total space of the cotangent bundle.