Parallel Transport Determines Connection

Theorem

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

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

Let $X$ and $Y$ be smooth vector fields on $M$ and $X_p$, $Y_p$ their values at $p \in M$.

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

$\map \gamma 0 = p$

$\map {\gamma'} 0 = X_p$

Let $\nabla_X Y$ be the covariant derivative of $Y$ along $X$.

Let $P^\gamma_{h_0 h_1}$ be the parallel transport map along $\gamma$.

Then:

$\ds \forall p \in M : \valueat{\nabla_X Y}p = \lim_{h \mathop \to 0} \frac {P^\gamma_{h0} Y_{\map \gamma h} - Y_p}{h}$

Proof

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Sources

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