Generalised connections over a vector bundle map

A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework.