## Curvature

Given a connection, parallel translation means propagating a tangent vector along the trace of a variation in such way that the resulting vector field along the curve is constant according to the connection. In the plane we can do much better than propagate a tangent vector merely along some curve. Indeed, we can propagate any given arrow over the whole plane to form a vector field whose arrows are all parallel translates of each other. In order to propagate a given arrow along a curve in a constant fashion, we need only form its field of parallel translates and restrict that to the curve in question. Hence the name parallel translation.