Transport along a curve

Definition

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 $v \in E_{\gamma(0)}$:

$\phi_0(v) = v$

and

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

Intuitively, we define a rule for moving the fibre of $E$, in a manner that is parallel to itself with respect to the connection.

Facts

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.