Suppose a curve on the unit sphere intersects every great circle on this sphere,and this curve has length $2\pi$,then prove that this curve must be the union of two halves great circles.
I know that a curve on the unit sphere with length less than $2\pi$ must miss one of great circle which is another way to say that such a curve must stay on hemisphere.The condition that this curve intersects every great circle means that this curve doesn't lie on hemisphere. But I don't know how to prove the final statement rigorously.