If you have a metric on $\mathbb{R}^2$ induced by the stereographic projection of $\mathbb{S}^2$ onto the tangent plane at its southern pole, what would a geodesic on that $\mathbb{R}^2$ plane look like?
We know that the stereographic projection is not isometric, but since we have an induced metric on the plane, the distances with respect to the induced metric are preserved, right?
And therefore the great circles, which are geodesics on the sphere, should be projected to geodesics on the plane. Is this correct? So geodesics on this plane would be arcs or circles. Is this correct?