TITLE: A walk on a compact 2D surface embedded in 3-space that never returns home
QUESTION [5 upvotes]: 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.)

REPLY [7 votes]: By Poincare recurrence theorem, on a compact manifold, almost every geodesic returns arbitrarily close to its starting point, infinitely many times.
So if you accept a slight error in the measure of the position (and direction) of the ant (say less than 1 picometer, and $10^{-9}$ degree), you are almost guaranted (w.r.t Lebesgue measure) that the ant returns to its initial position if you wait long enough.
The case of a non-compact manifold is even more interesting. 
If the volume is finite, there is always a geodesic that goes to infinity, yet the result still holds: a.e geodesics are recurrent.
There are also examples of infinite volume surfaces for which a.e. geodesics are recurrent. This is the case for a tubular neighborhood of the Cayley graph of $\mathbb{Z}^2\times \lbrace 0\rbrace$ in $\mathbb{R}^3$. 
Finally, there are examples of infinite volume surfaces for which there is a dense set of recurrent geodesics, yet almost all geodesics go to infinity. This is the case for a tubular neighborhood of the Cayley graph of $\mathbb{Z}^3$.