Connections on the Parallel Vector Fields in Vector Bundles

Abstract

This paper is a survey of the basic theory of connection on bundles. A generalised notion of connection on a fibre bundle  over a manifold  is presented. These connections are characterised by a smooth distribution on  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  into . Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map