I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of course). It was pointed out that the Chern-Gauss-Bonnet theorem shows that no such manifold exists in even dimensions. Also, it seems that by looking at the universal cover we see that no such manifold can be given both a spherical and hyperbolic, or both a spherical and euclidean structure. So, the only case that remains is showing that, in odd dimensions, no manifold can be given both a Euclidean and a hyperbolic structure.


Is there an example of a such a manifold, or a proof that no such manifold exists?



Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?
Here is another approach to impossibility. Some decades after Bieberbach, [Milnor](https://projecteuclid.org/euclid.jdg/1214501132) showed that the ball of radius $R$ in the universal cover of a compact manifold is basically a bunch of copies of fundamental domain, copies indexed by the ball of a similar radius in the fundamental group under the word metric. Thus a metric of negative curvature requires a fundamental group with exponential growth, while a flat metric requires a fundamental group with polynomial growth. These are not compatible, so no compact manifold can admit both metrics.


