is probably one of the hardest concepts of differential geometry
to grasp (I'm not sure if it really is, because I don't really understand it...)
. But here's a demonstration of how it works. You can also see it as a demonstration of holonomy
or the Gauss-Bonet theorem
, if you prefer those. The great thing about it is that it is equally impressive if you know nothing at all or a lot about differential geometry.
The demonstration shows how moving in straight lines on a sphere causes rotation! Note that this is emphatically not true on the plane. Here, "moving" means what is known as parallel transport, equivalent to moving your hand along a surface without turning it.
Let's start with the plane. Put your hand on the table, and trace it (without turning it!) along a triangle. Note how it comes back to rest at the same angle it started off with. Not very exciting...
Now let's try a sphere. Since you probably don't have one handy, just use your fully outstretched arm; your hand will rest on points of a sphere whose radius is the length of your arm. Start with your right hand by your thigh, palm facing inwards. Now move it forwards (keeping your arm stretched out, and not rotating your hand in any way!) 90o, till it is ahead of you. Now move your hand right (still with straight arm, not rotating anything) 90o right, till your arm is on the line of your shoulders. Finally, move your hand back 90o to your thigh, so your middle fingertip is back where it started. Remember! Don't rotate your hand through any of this, no matter how strange it feels!
If you've done it all correctly (and it's just 3 movements), your hand returned to its initial position, but your palm is now pointing forwards! Moving in a closed path along a sphere has rotated your hand!
For added effect, note that your hand described a triangle with angles summing to 270o degrees (this is possible on non-flat manifolds). The excess over the "correct" value of 180o, 90o, is exactly what your hand rotated by. If you make some of the angles in the triangle smaller, the equality of the excess and the rotation of your hand is preserved!