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I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of the metric (so there is no conjugate points on the universal covering as well).

Suppose that there exists two geodesics on the covering wich are strongly asymptotic on the future (the distance between them goes to zero as $t\rightarrow\infty$). Is there any hope of obtainning estimatives for the distance between them on the past (as $t\rightarrow -\infty$)?

I see that on the hyperbolic plane if the geodesics get closer on the future, they deviate on the past. I'm wondering if there is some similar "deviating behavior" (even if it was not monotonically increasing) in abscense of conjugate points too.

Thanks on advance.

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    $\begingroup$ Since Anton's answer is likely to disappear in the near future, here are some comments: 1. Giving a positive or negative answer for this question looks like a nice PhD thesis. 2. There are few positive ans some negative results in this direction, the most interesting counter-example is in Keith Burns' 1992 paper "The Flat Strip Theorem Fails for Surfaces with No Conjugate Points". 3. On the positive side, if you assume, in addition, that the metric has no focal points, strongly asymptotic geodesics will probably diverge in the opposite direction, see O'Sullivan's 1976 paper... $\endgroup$
    – Misha
    Commented Mar 23, 2013 at 13:34
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    $\begingroup$ "Riemannian Manifolds Without Focal Points". 4. Lastly, you should ask Keith Burns directly (he is still in Northwestern University) about the status of this question, since there were several followup paper since his 1992 work. $\endgroup$
    – Misha
    Commented Mar 23, 2013 at 13:37
  • $\begingroup$ Dear Anton, sorry for taking too long to unnacept the answer. I was really busy these days so I didn't see the changes on the status of the question. And thanks to Misha for pointing out the papers of Burns and Sullivan $\endgroup$
    – matgaio
    Commented Mar 24, 2013 at 21:07

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