Two Questions:
(1) Under what conditions(if any) can the logarithm map from a Riemannian manifold Q to it's Tangent Bundle TQ, locally, be a contraction mapping? For instance, over $\mathbb{R}$ it becomes a contraction mapping over the set $(1,\infty)$.

Or more generally, in terms of a Lipschitz constant,

(2) Given a flow on TQ and the canonical projection $\pi_Q$, if $\pi_Q\circ\Phi_h(q_0,v_0)=q_1$ and $\pi_Q\circ\Phi_h(q_0,\tilde{v}_0)=\tilde{q}_1$, then $exp(v_0)=q_0$ and $exp(\tilde{v}_0)=\tilde{q}_1$. Does there exist a constant depending on h, $C_h$, such that

$\|v_0-\tilde{v}_0\|=\|log(q_1)-log(\tilde{q}_1)\|\leq C_h\|q_1-\tilde{q}_1\|$