# Changes

## Transport along a curve

, 18:02, 6 January 2012
no edit summary
==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 a curve|connection along]] $\gamma$ of $E$. The transport along $\gamma$ defined by $D/dt$ maps $E_{\gamma(0)}$ to the space of [[section of a vector bundle along a curve|section]]s 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 [[frame]]s.