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 $\gamma <math>.Thetransportalong<math>\gamma <math>definedby<math>D/dt$ maps $T_{\gamma (0)}(M)$ to the space of vector fields along $\gamma <math>,denotedinsymbolsas:<math>v\mapsto {\varphi }_t(v)(t\in [0,1])$

such that for any vector $v\in T_{\gamma (0)}(M)$:

${\varphi }_0(v)=v$

and

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

Intuitively, we define a rule for moving the tangent space along the curve, in a manner that is parallel to itself with respect to the connection.