Transport along a curve
Let be a differential manifold, a vector bundle on . Let be a smooth curve in . Let denote a connection along of . The transport along defined by maps to the space of sections of along , denoted in symbols as:
such that for any vector :
Intuitively, we define a rule for moving the fibre of , in a manner that is parallel to itself with respect to the connection.
Connection gives transport
We saw above that a connection along a curve defines a rule of transport along that curve. Consider now a connection on the vector bundle. Then, for every curve, this gives rise to a connection along that curve, and hence a transport along that curve. Hence a connection gives a global rule for transporting frames.