I'm reading the following survey on algorithms for computing geodesic paths on discretized surfaces: http://www.cs.cmu.edu/~kmcrane/Projects/GeodesicSurvey/GeodesicSurvey.pdf
According to the authors, on the following diagram (on page 11 of the pdf) we can see paths from an arbitrary vertex $i$ to vertices $j$, a vertex of positive angle defect (think of angle defect as the discrete version of gaussian curvature), and $k$ a vertex of negative angle defect:
It's very clear to me why the exponential map fails to be injective when moving close to vertex $i$, the authors even describe themselves than the green and red paths converge on the other side (the boundary edges adjacent to $j$ in the diagram are glued together when this surface is embedded in $\mathbb R^3$)
However I'm struggling to understand why they claim that the exponential map fails to be injective when rays pass close to vertex $k$. They seem to split up and not converge later so I don't see why it can't be injective