I am looking for a geometry that is different from Euclidean geometry in that if the shortest path from A to B passes through C, we cannot necessarily conclude that the segment of the path from A to C is the shortest path to C. I would appreciate it if you could also recommend a good resource for...

I suggest deleting the tag decomposition-theorem. I would guess that the tag was intended to be about the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber. It seems unnecessary to have a whole tag just dedicated to this theorem (perverse-sheaves should suffice). What's worse, most po...