Parallel Transport in the Surface of a Cube


When we squash a sphere into a cube, we cannot remove its curvature in accordance with the Gauss-Bonnet theorem. The curvature gets pushed into the vertices, as each face is flat and we can move across edges with no effect. By unfolding the cube into its net, (keeping in mind that sides with the same color are identified with one another) we can define the curvature at each vertex as the amount of space missing around it relative to flat space. It is easy from outside to see that the curvature of each vertex must be the missing angle of pi/2, but how would we measure this curvature if we lived inside of the cube?

How does adjusting the slider change the vector? Does it translate it or rotate it?
What happens as we move across the red edges (which are identified)?
How does the vector change from its original position if we rotate counter-clockwise? Clockwise?
Did the vector experience any rotation from its perspective as it traveled around the vertex?
Note that the complete path is actually a triangle, as it is constructed from 3 straight sides from the perspective inside the cube!
What do the interal angles of this triangle sum up to?