Parallel Transport on a 2-Sphere
In a manifold, vector bundles can be defined as parallel . This Demonstration shows how parallel transport approximately appears on a sphere (a 2-manifold). The idea behind parallel transport is that a vector can be transported about the geometric surface while remaining parallel to an affine connection, a geometrical object that connects two tangent spaces on the surface. This is shown as two vectors starting at the same point on the sphere, ; then one is transported along the geodesic (the shortest line on a surface connecting two points) from to , while the other vector is shown traveling from to , then from to . Now the formerly parallel vectors are perpendicular, having moved on different paths on the surface, but remaining parallel to their respective connections along the way.
 Wikipedia. "Parallel Transport." (Sept 15, 2012) en.wikipedia.org/wiki/Parallel_transport.