About the Killing-Hopf theorem

Question

Theorem- Let $M^{n}$ a complete Riemannian manifold with constant sectional curvature equal to $K$. Then the universal cover $\bar{M}$ of $M$ , with covering metric is isometric to $H^{n}$ if $K=-1$, $\mathbb{R}^{n}$, if $K=0$ and finally $\mathbb{S}^{n}$ if $K=1$.

What information about the manifold $M$ the theorem tell me? Once universal cover $\bar{M}$ to be isometric to one of the spacial forms.

Thanks

Answer 1

Here is one example. You may have heard about the 3-dimensional Poincare Conjecture (if $M$ is a compact, without boundary, simply-connected 3-dimensional manifold, then $M$ is homeomorphic to $S^3$). Here is how it was proven by Gregory Perelman, broadly speaking:

1. $M$ is known to admit a smooth structure. Equip $M$, therefore with a randomly chosen Riemannian metric $g$. Since we know nothing about the properties of $g$, we cannot deduce any conclusions about $M$ from the existence of $g$.

2. Deform $g$ via the Ricc flow (with surgeries) to an Einstein metric $g_0$ on $M$. (OK, not on $M$ itself but on pieces of a connected sum decomposition of $M$, but let's ignore this.)

3. In dimension 3 each Einstein metric has constant curvature. It is easy to see that this curvature has to be positive (since $M$ is compact and simply connected), hence, after rescaling, we can assume it to be equal to $1$. Hence, by the Killing-Hopf theorem (I always thought of it is Elie Cartan's theorem...) $(M, g_0)$ is isometric to the unit 3-dimensional sphere with its standard metric. In particular, $M$ itself is diffeomorphic to $S^3$.

Now, you see how this theorem is useful and what advantage one has having a metric of constant curvature on a manifold.

Answer 2

For me, the Killing-Hopf theorem is interesting because it tells you that a complete connected Riemannian $n$-manifold $M$ with constant non-positive sectional curvature is a classifying space $B\Gamma$ (i.e. an Eilenberg-MacLane space $K(\Gamma,1)$) for its fundamental group $\Gamma$. This discrete group is a subgroup of the isometry group of the universal cover.

This follows because the universal cover $\widetilde{M}$ is contractible in the non-positive sectional curvature case, so by the long exact sequence of homotopy groups, we have isomorphisms $\pi_k(M) \cong \pi_k(\widetilde{M})$ for $k \geq 2$.

From the point of view of geometric group theory, this is interesting because it means that cohomology of $M$ with local coefficients is isomorphic to group cohomology of the fundamental group $\Gamma$.