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:


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