Nonpositive curvature, Theorem of Cartan-Hadamard In my differential geometry course we had the following
Theorem (Cartan-Hadamard): Let $M$ be a connected, simply connected, complete Riemannian manifold. Then the following are equivalent:


*

*$M$ has nonpositive curvature

*$|d\exp_p(v)\hat v| \ge |\hat v|$ for all $p \in M$, $v, \hat v \in T_pM$

*$d(\exp_p(v), \exp_p(w)) \ge |v - w|$ for all $p \in M$, $v, w \in T_pM$


In addition $\exp_p$ is a diffeomorphism if either of these statements holds.
Now, I understand the proof that was given, but I do not see where we have to use simple connectivity of $M$ (in the proof given by my professor the Cartan-Ambrose-Hicks Theorem was used). What's wrong with the following:
$1. \Leftrightarrow 2.$ didn't make use of simple connectivity. (at least I don't see where)
But now I think 2., completeness and connectedness imply that $\exp_p$ is a diffeo, and in particular $M$ is simply connected.
_ 2. implies 3.: Let $v, w \in T_pM$ be given, let $\gamma(t) = \exp_p(v(t))$ be a lenght minimizing geodesic connecting $\exp_p(v)$ and $\exp_p(w)$. Then
$
\begin{eqnarray}
 d(\exp_p(v), \exp_p(w)) &=& L(\gamma) \\ &=& \int_0^1 |\dot \gamma(t)| \, dt \\ &\ge&  \int_0^1 |\dot v(t)| \, dt \\ &\ge&  \left| \int_0^1 \dot v(t) \, dt \right| \\ &=& |v-w|
\end{eqnarray}
$
Therefore $\exp_p$ is injective. 
Completeness and connectedness imply that $\exp_p$ is surjective (Hopf-Rinow).
2 implies that $\exp_p$ is a local diffeo.
So in conclusion $\exp_p$ is seen to be a bijective local diffeo, hence a diffeo.
My question: 


*

*Is there anything wrong with this argument? 

*If yes: Could you give an example of a complete, connected manifold with nonpositive curvature, which is not simply connected?


Thanks a lot!
S.L.
 A: Regarding your comment above: yes, if $M$ is complete and connected with non-postiive curvature, then the universal cover $\widetilde{M}$ of $M$ inherits 
a complete connected non-positive curvature metric (just pull-back the Riemannian
metric from $M$), and so satisfies the conditions (and so also the conclusions!) of the Cartan--Hadamard theorem.  One then finds that $\widetilde{M}$ is homeomorphic to the tangent space of any point $\widetilde{m}$ of $\widetilde{M}$ (via $\exp_{\widetilde{m}}$.  If $m$ is the image of $\widetilde{m}$ in $M$, then
the tangent space to $\widetilde{m}$ is naturally identified with the tangent space to $m$, and the exponential map $\exp_m$ is naturally identified with the composite of  $\exp_{\widetilde{m}}$ and the projection $\widetilde{M} \to M$.
Thus indeed, one finds that $\exp_m$ is a covering map.  
This circle of ideas is frequently applied, e.g. in the context of hyperbolic manifolds.  If $M$ is a compact connected hyperbolic manifold, then one finds 
by Cartan--Hadamard that $\widetilde{M}$ is isometric to hyperbolic space of 
the appropriate dimension, and so $M$ is isometric to a quotient of hyperbolic space by a discrete cocompact group of isometries.  (This is why hyperbolic manifold theory interacts so tightly with certain parts of group theory.)
Also, one concludes that if $M$ is compact and connected (and positive dimensional --- i.e. not a single point!) with non-positive curvature then $\pi_1(M)$ is infinite (because, writing $M$ as a quotient of $T_m M$ via
$\exp_m$, we see that to get something compact, we have to quotient out by an
infinite group of diffeos).
