Definition:Parallel Transport Map

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Let $I \subseteq \R$ be an open real interval.

Let $M$ be a smooth manifold.

Let $\gamma : I \to M$ be a smooth curve.

Let $T_p M$ be the tangent space of $M$ at $p \in M$.

Let $v \in T_{\map \gamma {t_0} } M$.

Let $V$ be the parallel transport of $v$ along $\gamma$.

Then the parallel transport map is defined as the map:

$P_{t_0 t_1}^\gamma : T_{\map \gamma {t_0} } M \to T_{\map \gamma {t_1} } M$

such that:

$\forall v \in T_{\map \gamma {t_0} } M : \map {P_{t_0 t_1}^\gamma} v = \map V {t_1}$