This is by far the best answer and it comes with a reference, so THANK YOU VERY MUCH. I'll take a look at that book if I can get my hands on it.
On 22/01/2014 01:17, Leon Aigret wrote: > On Sun, 19 Jan 2014 17:40:23 +0100, ?XPOSITO <firstname.lastname@example.org> > wrote: > >> Can somebody provide an example of a connected surface containing **NO** >> non-plane geodesics, >> apart from those surfaces contained in the plane or the sphere? > > Probably not. Theorem 6.7.1 at the bottom of page 74 in Differential > and Riemannian Geometry by Detlev Laugwitz states: > > If all geodesics of a surface are plane curves then the surface is a > piece of a sphere or of the plane. > > Leon >