# the shortest path between two points and the unit sphere and the arc of the great circle

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

Thanks

• I'm assuming we are to use the usual Riemannian metric. Is this correct? – Robin Goodfellow Sep 17 '14 at 2:42
• We are using Crafton's Formula – user176127 Sep 17 '14 at 3:32
• 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! – 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.