Homogeneous nonlinear splittings and Finsler submersions

A nonlinear splitting on a fiber bundle is a generalization of an Ehresmann connection. An example is given by the homogeneous nonlinear splitting of a Finsler function on the total manifold of a fiber bundle. We show how homogeneous nonlinear splittings and nonlinear lifts can be used to construct submersions between Euclidean, Minkowski and Finsler spaces. As an application we consider a semisimple Lie algebra and use our methods to give new examples of Finsler functions on a reductive homogeneous space.