On the unit sphere $S^2$, show that it is not true that the a geodesic curve is the shortest curve between any two points.
The geodesic curves on $S^2$ are great circles. So we can choose an arc $\gamma$ on a great circle such that $\pi<\text{length}(\gamma)<2\pi$, then $\gamma$ is not the shortest curve between two endpoints....
Am I right? Are there any rigorous proofs? Thanks in advance!