Fary's theorem states that any simple planar graph can be drawn without crossings so that its edges are straight line segments (see Wikipedia). The proof is based on the Art gallery theorem, so I would have asked also this latter. But, perhaps there is a simpler answer on my question.
In any case, my question is the following.
Is it true, that any simple planar graph can be drawn on the sphere without crossings so that its edges are geodesic segments?