Connections on a Principal Bundle

Abstract

This chapter discusses connections on a principal bundle. Throughout the chapter, G will be a Lie group with Lie algebra g. One possible definition of a connection on a principal G-bundle P is a C∞ right-invariant horizontal distribution on P. Equivalently, a connection on P can be given by a right-equivariant g-valued 1-form on P that is the identity on vertical vectors. The chapter shows the equivalence of these two definitions of a connection. A connection is one of the most basic notions of differential geometry. It is essentially a way of differentiating sections. From a connection, the notions of curvature and geodesics follow.