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? 


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?



The logarith map as a contraction
I interpret the question to be intuitively


the map of a small piece of a manifold back to its tangent distorts distances


expanding for positive curvature and contracting for negative curvature 


and neither for zero curvature


so the answer intuitively sought is : for negatively curved spaces the inverse of the exp map is contracting


dennis sullivan


