I'm listening to Prof. Leonard Susskind's lectures on GR on youtube.com at http://www.youtube.com/watch?v=hbmf0bB38h0&feature=relmfu He's trying to explain how to visualize parallel transport of a vector. But I'm having a hard time of it. I think I understand it. Let me know if I got it straight: The axes of a coordinate systems can change as you move along a line... with respect to the prior coordinate system. In other words, the basis vectors can smoothly change direction along a line. And the parallel transport of a vector ensures that the direction of a vectors follows the changes in the direction of the basis vectors as the orientation of the coordinate system changes along the line. Or, parallel transport ensures that the vector is always in the same direction with respect to the local coordinate system no matter where you are on a line. Is this right?