At the risk of asking an uninformed question...


Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once placed and oriented, the ant will walk along a local geodesic path.


Are there examples of such 2D surfaces where we are guaranteed the ant will never return to some starting position? 


(Thanks to Henry Wilton for requesting clarification as per his comment below.)



A walk on a compact 2D surface embedded in 3-space that never returns home
Silly answer: any plane in 3-space will work (so the answer is "yes").


You probably want a compact surface. These always admit geodesic cycles (so the answer in this case is "no").


