We can visualize as with pairs of antipodal points identified together. Let’s assign a forward and right direction at one point (notice that when we assign direction, the same direction is assigned to the two antipodal points) and we what goes wrong.
Initial positionMoving in the forward direction on the sphere and assigning a rightNotice when the two paths meet, the forward direction gets messed up.
Hence, orientation is not well-defined.
This won’t happen on :
Initial position with chosen directionMoving forward along the circleWhen we meet, the choices do not conflict.
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