Connections

Abstract

In this chapter we show that by introducing a new object, called a connection, in a manifold, we are able to translate a tangent vector at a point of the manifold to another point, if these two points can be connected by means of a curve on the manifold. This is equivalent to be able to calculate directional derivatives of vector fields. As we shall see in Sects. 6.2 and 7.4, there is a connection defined in a natural manner if the manifold has a Riemannian structure or is a Lie group. The properties imposed on a connection have their origin in the study of two-dimensional surfaces in the three-dimensional Euclidean space and lead to the concept of curvature.