Hi all Making this title was harder than I thought. It certainly makes the topic look more advanced than it actually is. I studied differential geometry during my masters but never went much in depth, just enough so I could apply basic concepts to my specific problems at the time. Now I'm trying to answer a simple question: do parallel transport on closed loops on curved spaces conserve scale? scale as in volume. Also, the question is specific to 2D spaces, or surfaces, so scale would be the area of a polygon. So bear with me because my knowledge on the subject is very limited in solid math and too reliant on my intuition. I spent a long time dismissing this question in my head because my intuition was making some assumptions that I now question. Clearly the area of an object can change, (with respect to a projection on an euclidean surface) but if I take this object for a walk around space and come back to the initial position, its orientation might change but its scale doesn't seem like it would. Unless there is some kind of discontinuity, or holes, on this space or I impose specific boundary conditions that connect edges of different sizes. Which makes me wonder if this question is more suitable for the topology forum than differential geometry, because holes and connections scream topology to me. Does this make sense?