Bactra Review    Geometrical Methods of Mathematical Physics
Consider a unit vector at the North Pole, pointing towards the 0-degree line, the prime meridian. Move it forward along the prime meridian, keeping it tangent to the sphere --- which certainly looks like parallel transport. Keep following the prime meridian until you come to the South Pole; the vector will now be pointing towards the 180-degree line. If, again starting from the North Pole, you begin moving the vector south along the 90-degree line, keeping it tangent to the sphere and perpendicular to the meridian --- which also looks like parallel transport --- the vector arrives at the South Pole pointing towards the prime meridian. So which of the two opposite vectors at the South Pole is really parallel to the original one at the North Pole? (This example is shamelessly stolen from p. 202 of Schutz.)