The purely affine notion of vectors and their duals is not enough to define the length of a vector in general; it is only sufficient to define a length relative to other lengths along the same geodesi...The purely affine notion of vectors and their duals is not enough to define the length of a vector in general; it is only sufficient to define a length relative to other lengths along the same geodesic. When vectors lie along different geodesics, we need to be able to specify the additional conversion factor that allows us to compare one to the other. The piece of machinery that allows us to do this is called a metric.
