No twisting and no lifting is parallel transport — the Levi-Civita connection on the sphere. To draw the
unfolding, each leg of the path is placed on the developable surface tangent to the sphere along that arc:
a great-circle (geodesic) segment lies on a tangent cylinder and flattens to a straight segment in the plane;
a small-circle segment lies on a tangent cone and flattens to a circular arc when the cone sector is unrolled.
Trace the path on the matching cylinder or cone between consecutive corners, slit, and lay the sheet flat so
tangents stay coherent along the cut; the angle by which a closed loop fails to close in the plane is
holonomy.