## Connections

Recall that a vector field is a choice at each point of a manifold of a vector in the tangent space at that point. The concept of the rate of change of a vector field has a profound relationship to geometry. This is because a straight line is a curve traced out by a point moving with constant velocity. The succession of velocities which a moving point adopts constitutes a vector field along the curve traced out during the course of its motion. We call this vector field the velocity field along the curve. The velocity field of a constant velocity motion is a constant vector field, and so these particular vector fields must be considered to have rate of change zero. If one considers the possibility of different geometries on a plane, so that the straight lines in these new geometries are not the same as the usual Euclidean straight lines, then our idea of a constant velocity motion must be changed to one that will trace out the new kind of straight line. Consequently the whole specification of the rate of change of vector fields must alter, since even the basic idea of what has rate of change zero must be tailored to produce the different families of straight lines.