Two Questions: (1) Under what conditions(if any) can the logarithm map from a point on a Riemannian manifold, $q_1\in Q$, to the Tangent Space $T_{q_0}Q$, locally, be a contraction mapping? Intuitively, I'm thinking of the logarithm function over $\mathbb{R}$ being a contraction mapping over the domain $(1,\infty)$, and I'm interested in seeing if an analogous situation can occur for a Riemannian manifold, when we are restricted to a local neighbourhood of $q_0$.

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

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

if we restrict ourselves to a local neighbourhood of $q_0$, so that the exponential map is injective?

If the exponential map is defined locally and the domain is shrunk it becomes arbitrarily close to an isometry. Is there a way to measure how close it is to an isometry given the domain has been shrunk to a particular size?