I'm starting to learn Riemannian geometry and have a question.
Let $\mathcal{M}$ be a Riemannian manifold, $p \in \mathcal{M}$; $\tau_{p}^{q}$ be a parallel transport from $p$ to $q$ and $\operatorname{exp}_p$ be an exponential map. The question is: given a smooth curve $\gamma : [a, b] \to \mathcal{M}$ such that $\gamma([a, b]) \subset \mathcal{U}$, where $\mathcal{U}$ is some neighborhood of $p$ where $\operatorname{exp}_p, \tau_{p}^{(\cdot)}$, are defined, is it true that \begin{equation} \left( \frac{\mathrm{d}}{\mathrm{d}t} \operatorname{exp}_p^{-1}\gamma \right)(t) = \tau_{\gamma(t)}^{p}\dot{\gamma}(t) \,? \end{equation}