Transport along a curve

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Let M be a differential manifold, E a vector bundle on M. Let \gamma:[0,1] \to M be a smooth curve in M. Let D/dt denote a connection along \gamma of E. The transport along \gamma defined by D/dt maps E_{\gamma(0)} to the space of sections of E along \gamma, denoted in symbols as:

v \mapsto \phi_t(v) (t \in [0,1])

such that for any vector v \in E_{\gamma(0)}:

\phi_0(v) = v


\frac{D (\phi_t(v))}{dt} = 0

Intuitively, we define a rule for moving the fibre of E, 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.