There is a theorem of Cartan which states that if $M$ is a simply connected, complete Riemann manifold, and that the sectional curvature is everywhere $\leq 0$, then any two points of M are joined by a unique geodesic.

So if we consider a Riemann manifold $M$ which is not simply connected, but is complete and has sectional curvature $\leq 0$, we could use the above theorem of Cartan to the universal covering space $\widetilde{M}$ of $M$. It says in the book:

For it is clear that $\widetilde{M}$ inherits a Riemannian metric from $M$ which is geodesically complete, and has sectioanl curvature $\leq 0$.

Given two points $p,q \in M$, it follows that each homotopy class of paths from p to q contains precisely one geodesic.

My question is: How does the second sectence deduced from the sentence above? I know that any two points of $\widetilde{M}$ are joined by only one geodesic, but for any two points $p,q \in M$, there are many lifted points of $p,q$ in $\widetilde{M}$. Thank you!

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    $\begingroup$ Pick $p$ as your basepoint. Consider that each homotopy class of maps from $p$ to $q$ lifts to a unique point in $\widetilde{M}$. Consider the geodesic connecting the constant path at $p$ (an element of $\widetilde{M}$) to this point. $\endgroup$ Jul 15, 2012 at 2:49
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    $\begingroup$ What book would that be? $\endgroup$
    – Will Jagy
    Jul 15, 2012 at 2:49
  • $\begingroup$ @WillJagy The book is Morse Theory by Milnor $\endgroup$
    – Proton
    Jul 15, 2012 at 5:08
  • $\begingroup$ @Anonymous Ah,thank you! I forgot the unique lifting theorem. $\endgroup$
    – Proton
    Jul 15, 2012 at 5:21
  • $\begingroup$ @AnonymousCoward : Consider posting your comment as an answer so that other differential geometers won't think that this is an unanswered question. :) $\endgroup$
    – treble
    Jul 16, 2012 at 6:35

1 Answer 1


This answer is by AnonymousCoward from the comments. I am posting it as an answer for completeness' sake.

We have $M$ a non-simply connected complete Riemannian manifold with nonpositive sectional curvature. Pick $p$ as the basepoint of $M$. If $q\in M$, then any homotopy class of curves $\gamma$ connecting $p$ and $q$ determine a unique lift of $q$ to $\widetilde{M}$. (Observe that in particular the nullhomotopic paths determine a lift of $p$.) Every curve in $\gamma$ lifts to a curve connecting the lifts of $p$ and $q$, and conversely every curve in $\gamma$ is the projection of a curve connecting $p$ and $q$.

Since the metric on $M$ pulls back to $\widetilde{M}$, we apply Cartan's theorem to the lifts of $p$ and $q$ and find there is a unique geodesic in $\widetilde{M}$ connecting the lifts of $p$ and $q$. Since the universal covering map is a local isometry, the image of this geodesic is again a geodesic in $M$. Uniqueness gives that it is unique in its homotopy class.


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