  While in Riemannian geometry the parallel transport of length is  path independent, that is, it is possible to compare the  lengths of any two vectors, even if they are located at two finitely  different points, a vector suffers a path-dependent change  in direction under parallel transport; that is, it is not  possible to define the angle between two vectors, located at  different points, in a path-independent way. Consequently, the angle  between two vectors at a given point is invariant under parallel  transport if and only if both vectors are transported along the  same path. In particular, a vector which is carried around a  closed circuit by a continual parallel displacement back to the  starting point, will have the same length, but will not in general  return to its initial direction. 