Connections

Abstract

The tangent space, T _x M, to a differentiable manifold M at a point x is a vector space different from the tangent space to M at any other point y, T _y M. In general, there is no natural way of relating T _x M with T _y M if $x \not= y$. This means that if v and w are two tangent vectors to M at two different points, e.g., v∈T _x M and w∈T _y M, there is no natural way to compare or to combine them. However, in many cases it will be possible to define the parallel transport of a tangent vector from one point to another point of the manifold along a curve. Once this concept has been defined, it will be possible to determine the directional derivatives of any vector field on M; conversely, if we know the directional derivatives of an arbitrary vector field, the parallel transport of a vector along any curve in M is determined.