Suppose $(M,g)$ is a Riemannian manifold of curvature $K$ where $k\leq K\leq 0$, for some negative number $k$. Is there an explicit formula describing the difference between
- $ d_g(x,y) $ and
- $ \|Log(x,x_0)-Log(y,x_0)\|, $
- $d_H(Exp^H(z,Log(x,x_0)),Exp^H(z,Log(y,x_0)))$
where $x,y,x_0\in M$, Log is the Riemannian Log map on $M$, $Exp^H$ is the Riemannian exp map on the hyperbolic space $H^d$of same dimension as $M$, $z$ is an element of $H^d$ and $d_g$, $d_H$ is the metric induced by $g$ and the hyperbolic metric respectively?