A walk on a compact 2D surface embedded in 3-space that never returns home 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: 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 $\mathbb{Z}^3$.
A: 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").
A: Every smooth two-manifold with nonzero genus has a locally geodesic closed loop: the smallest noncontractible cycle. But that doesn't rule out the existence of a point that is not on a geodesic loop, nor does it rule out the possibility that the closed loops through a point have measure zero among the random choices of direction.
If you just want a single surface, a single point on that surface, and probability one of not returning home, take a standard torus in 3d. If you pick any point on that surface, and choose an orientation randomly, you will (with probability one) get a curve such that it has an irrational change in longitude when it first returns to the same latitude as its starting point.
A: An easy case is the flat torus (that is, a torus with the flat metric -- the one induced from its representation as a square in the plane with edges identified).  There, the ant will return home if and only if the slope of its walk is rational.  I think this means that with probability 1, the ant will not return home!
A: A flat torus doesn't work because it can't be embedded into 3-space (at least, I'm assuming by "3-space" you mean R^3, not, say, T^3, in which case it can be embedded).
