Let $(W,S)$ be a Coxeter system and let $\Sigma$ be the corresponding Davis complex. It is well-known that the Davis complex may be equipped with a piecewise Euclidean metric so that it is a proper, complete $\text{CAT}(0)$ metric space. Being the 1-sceleton of the Davis complex, the Cayley graph of $W$ (with respect to the generating set $S$) canonically embeds into $\Sigma$.
It might be obvious, but I was wondering if this embedding sends geodesic paths (with respect to the word metric) in the Cayley graph to geodesic paths in $\Sigma$ (with respect to the piecewise Euclidean metric).
It does not send geodesic paths to geodesic paths. For example in the finite dihedral group case the complex is a polygon and the Cayley graph is on the boundary.