# How does the Cartan-Hadamard theorem demonstrate the impact of non-positive sectional curvature on the topology of a manifold?

My note says that Cartan-Hadamard theorem is an example of how negative sectional curvature "impacts" the topology of manifolds. The version of the theorem we are using is:

Let $(M,g)$ be a complete Riemannian manifold with everywhere non-positive sectional curvature. Then $\exp_p: T_pM \to M$ is a covering map.

I don`t understand what it means to "impact" the topology. How does the theorem exhibit such an effect?

• For example, if $M$ is simply connected, then a corollary says that exp$_p$ is a diffeomorphism, hence $M \cong \mathbb{R}^m$. In particular, if you don't want $M$ to be diffeomorphic to $\mathbb{R}^m$, then $M$ cannot be simply connected. Aug 13, 2016 at 13:58
• For (another) example, the theorem implies that $M$ is covered by $\mathbb R^n$, which means that all the higher homotopy groups $\pi_n(M)$ are trivial. This exclude a lot of manifolds.
– user99914
Aug 13, 2016 at 17:13
• the question is what does it mean to "impact" the topology...
– user166467
Aug 15, 2016 at 6:56

It would have helped if you were to include a complete quote and its source. As written, the quote (and, thus, your question) can be interpreted in two different ways:

1. How does C-H theorem impacts "topology of manifolds", the latter regarded as an area of mathematics, with thousands of papers written in the last 100+ years, famous conjectures solved (like the Poincare conjecture), etc.

2. More narrowly: If we know that a smooth manifold $M$ admits a complete Riemannian metric of negative curvature, what does C-H theorem say about the topology of the manifold $M$?

The universal cover of $M$ is diffeomorphic to $R^n$. In particular, $\pi_i(M)=0$, $i>1$ (i.e. $M$ is aspherical), $\pi_1(M)$ has no nontrivial elements of finite order. There are other, more subtle implications, but that's already enough to get an idea, since these restrictions exclude many manifolds.
For the question in the 1st interpretation, the answer is much more ambiguous and depends on whom you ask. From my viewpoint, C-H theorem provides a source examples of manifolds $M$ whose universal covers are diffeomorphic to $R^n$ and, hence, aspherical manifolds. Study of aspherical manifolds is an important part the modern algebraic topology with the central questions revolving around conjectures of Borel and Wall. For instance, Borel conjecture states that two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic. While this conjecture is currently out of reach, it was proven (in dimensions $\ne 4$) when one of the manifolds admits a metric of nonpositive curvature, see