Given a Riemannian manifold $\mathcal{M}$, $p,q\in \mathcal{M}$, $\log_p:\mathcal{M}\rightarrow T_p\mathcal{M}$ denote the logarithm maps. Defining its derivative: $$d[\log_pq]:T_q\mathcal{M} \rightarrow T_p \mathcal{M}$$ I want to know whether the derivative of the logarithm map is isometric? i.e., $$\|d[\log_pq] (\xi)\|_p = \|\xi\|_q,~~ \forall \xi \in T_q \mathcal{M}$$ Or under what conditions can it be established?
In addition, what connection between parallel transport and derivative of the logarithm map?