31/10/2026
Imagine carrying an arrow around the surface of a sphere, carefully keeping it from turning as you move. Surprisingly, when you return to where you started, the arrow may point in a completely different direction.
This is a consequence of curved geometry known as parallel transport. In this visualization, a vector travels along a triangular path on a sphere. Although its direction is kept locally parallel throughout the journey, the curvature of the surface causes its final orientation to differ from its initial one. The amount of rotation depends on the area enclosed by the path and the sphere's curvature.
This simple geometric effect reveals something fundamental about curved space. Even without directly measuring its shape, we can detect curvature by transporting a direction around a closed loop and observing how it changes.