Definition:Parallel Transport Map
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Definition
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:
- $\forall v \in T_{\map \gamma {t_0} } M : \map {P_{t_0 t_1}^\gamma} v = \map V {t_1}$
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Parallel Transport