Parallel Transport Determines Covariant Differentiation

Theorem

Let $M$ be a smooth manifold with or without boundary.

Let $TM$ be the tangent bundle of $M$.

Let $\nabla$ be a connection in $TM$.

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

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

Let $P_{t_1 t_0}^\gamma$ be the parallel transport map along $\gamma$.

Let $D_t$ be the covariant derivative along $\gamma$.

Suppose that $V$ is a smooth vector field along $\gamma$.

Then:

$\ds \forall t_0 \in I : \map {D_t V} {t_0} = \lim_{t_1 \to t_0} \frac {P^\gamma_{t_1 t_0} \map V {t_1} - \map V {t_0} }{t_1 - t_0}$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Parallel Transport