I've read Collier's book on General Relativity and consulted parts of Schutz, Hartle and Carroll. In the terms they use, i have yet to gain anything resembling an intuitive understanding of parallel transport. In fact, it seems to me it is usually presented backwards, saying that the geodesic is the path where the covariant derivative of the tangent vector to the path is zero. This is true, but it seems to me simpler to say that the geodesic is chosen parallel to the tangent vector whose covariant derivative is zero. Let me explain in really simple terms. I was trying to visualize it as I walked my dog thru the local park this morning. The park is on a gentle slope with hillocks here and there, so that it is a nice 2-D manifold with a tangent space at each point. The path starts out on level ground and is so just a straight line in the direction of my destination, a point at the top of the slope where there is a gate. So the tangent vector to the path parallel transports, i.e., it remains constant, its derivative, zero. Now I reach a spot where there is a hillock nearby, ahead on the left. Suddenly, I am not in flat space any more and I must take that into account. What do I do? I first adjust my tangent space to the terrain, so it is a plane tilted slightly with respect to the preceding one, meaning the coordinate basis has changed. I then pick the vector in the space which is approximately parallel to the preceding one and whose covariant derivative is zero. And I make my path tangent to that vector, i.e., I adopt that vector as the new tangent vector to my path, effectively choosing the path. I then advance a distance dx and start over at the beginning of the last paragraph. Is this reasonable or have I entirely missed something -- hopefully simpler? I have seen another (closed) discussion on this topic which says a number of things which mean nothing to me: "Parallel transport just keeps the vector at the same angle it started, irrespective of the worldline that's transporting it" OK, parallel with respect to what? "...parallel transport is defined with respect to the connection...?" I did not realize a connection was a set of coordinates. I thought it was a function of derivatives of the metric, commonly known as a connection coefficient or Christoffel symbol. I guess my questions indicate the degree of my confusion. I need some intuitive understanding of this. It is not adequate to me to say that covariant derivate = zero defines parallel transfer. Sorry for the confusion and thanks in advance for any help.