conditions for long geodesics without self-intersections Consider a Riemannian manifold $\Sigma$  of dimension two homeomorphic to a torus. When is there a non-closed geodesic on $\Sigma$ which does not intersect itself — are there reasonable necessary or sufficient conditions for this? 
(This is a reference request. My guess is that this question does not have an amazing answer, but I am curious if it was considered in the literature. By the way, a flat torus obviously has this property, but I am interested in a more general setting.)  
 A: This is more of a comment intended to provide examples.  One can construct a reasonably large class of examples of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics (described below) is stable with respect to small $C^{\infty}$ perturbation. 
Infinite geodesics on surfaces without self intersections are closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp  close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)
https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps

          


          

Fig.11. A geodesic lamination on the punctured torus.


Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.

          


I believe that both examples are stable under small perturbation of the metric.
