Prove that the shortest path between two points on the unit sphere is an arc of a great circle connecting them

Great Circle: the equator or any circle obtained from the equator by rotating further: latitude lines are not the great circle except the equator

I need help with starting this question, because I am not quite sure how to prove this

Any help will be highly appreciated


  • $\begingroup$ I'm assuming we are to use the usual Riemannian metric. Is this correct? $\endgroup$ – Robin Goodfellow Sep 17 '14 at 2:42
  • $\begingroup$ We are using Crafton's Formula $\endgroup$ – user176127 Sep 17 '14 at 3:32
  • $\begingroup$ It would be good to include the Crofton's Formula approach in your original post, as we would never have guessed. It's a very nice problem, though! $\endgroup$ – Ted Shifrin Sep 17 '14 at 3:52

HINT: Start with two points on the equator. Every great circle (except one) meets the shorter great circle arc joining them in one point. Show that for any other curve $C$ joining the points, there must be an open set of great circles that meet $C$ in at least two points.


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