I recently read many articles in the context of the AdS/CFT correspondance in which the geodesic approximation is used (see for example section 3.5 here). The correlator between two boundary operators with scaling index of $\Delta$ and the geodesic length $d$ of a bulk field of mass $m$, with $\Delta=\frac{d}{2}+\sqrt{\frac{d}{2}+m^2}$ (the radius of AdS is one), is given by $$ <O(x)O(y)> \sim e^{- m d(x,y)} $$ for $m \gg 1$. $d(x,y)$ represents the geodesic distance between points $x$ and $y$.
I have no problem with this, what I didn't understand is why this is true only for geometries which can be analytically continued to Euclidean signature. Why is this, or more generically, where is my geodesic approximation coming from?
