# The number of geodesics of a complete Riemann manifold with non-positive sectional curvature

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!

• 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. Jul 15, 2012 at 2:49
• What book would that be? Jul 15, 2012 at 2:49
• @WillJagy The book is Morse Theory by Milnor Jul 15, 2012 at 5:08
• @Anonymous Ah,thank you! I forgot the unique lifting theorem. Jul 15, 2012 at 5:21
• @AnonymousCoward : Consider posting your comment as an answer so that other differential geometers won't think that this is an unanswered question. :) Jul 16, 2012 at 6:35

## 1 Answer

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.