I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here.
I'm trying to understand some aspects of the geodesics of compact surfaces without conjugate points and the following question came out: consider the 2-torus with negative constant curvature, its universal covering $\mathbb{R}^{2}$ with the covering metric and two geodesics in the covering wich are asymptotic to each other in the "future". It is true that the distance between these two geodesic increases monotonically in the "past", because of the negative curvature (see the figure).
I'm trying to see if this is true for universal coverings of compact surfaces without conjugate points. To be more precise, consider two geodesics wich are asymptotic to each other on the "future" on the universal coverig of a compact surface (2-torus, 3 torus etc). Is it true that the distance between them does not decrease on the past? I was thinking the following: if it was false, something like the following should happen:
Perhaps it is possible to use some "shortcut" argument to prove this is impossible (if the result I want is true...) but I'm not able to construct "the smart shortcut", perhaps using the fact that the "upper" geodesic approaches a little bit to the "bottom" geodesic before the local maximum as we see in the figure. My idea lies on the fact that the "no conjugate points" conditions implies that on the universal covering every geodesic is minimizing. The fact that in the universal covering we have some control of the "periodicity" of the metric could give some hint too.
Thanks a lot on advance!
firstly I just want to point out that the universal cover of the two holed torus (and in fact of the $n$-holed torus where $n>1$) is not $\mathbb{R}^2$ but the upper half plane (I'm going to denote this as $\mathcal{H}$) equipped with the Poincare metric: $g_{x+iy} = \frac{dxdy}{y^2}$.
Your first picture is a little misleading, since the geodesics on the upper half plane are precisely vertical lines and semi-circles with centre on the real axis. If we try and draw two geodesics which asymptote towards each other, say the semicircle centred at $0$ and of radius $1$ and the vertical line at $1$, we see that although they appear to asymptote towards each other as we approach the point $1$, the metric is actually blows up as $y\rightarrow 0$, so the actual distance between these two geodesics is heading to infinity. So, I don't think that there actually exists a pair of geodesics on $\mathcal{H}$ which asymptote towards each other.