# Difference between revisions of "Transport along a curve"

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## 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 $\gamma[itex]. The transport along [itex]\gamma$ defined by $D/dt$ maps $T_{\gamma(0)}(M)$ to the space of vector fields along $\gamma[itex], denoted in symbols as: [itex]v \mapsto \phi_t(v) (t \in [0,1])$

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

$\phi_0(v) = v$

and

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

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.