Fibred spaces and non-linear connections

Summary We give the fibre structure to the two complementary distributions, one being holomonic and the other being in general non-holomonic, by assigning the structure preserving transformations of the coordinates and of the frames. By the conversion from such fibred spaces we derive the spaces of non-linear connections as their image, in which we point out that the usual conditions imposed on C_kji and the covariant differentiation of g_ji in the metric spaces of non-linear connections can be interpreted respectively as the parallel and isometric conditions imposed on the fibre of the corresponding fibred Riemannian spaces. To Enrico Bompiani on his scientific Jubilee.