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 $\mathbb{Z}^3$.