Good morning everyone,


I was wondering about the difference between manifolds carrying a Riemannian metric with negative sectional curvature and hyperbolic manifolds. I was told once "there are very few properties in constant negative sectional curvature that cannot be extended to the negative sectional curvature case".


Can one list the topological properties a manifold with negative (but variable) sectional curvature must have?



Negative sectional curvature and constant curvature
1. Sullivan proved that every closed hyperbolic manifold has a stably parallelizable finite cover. This is not true for say complex hyperbolic manifolds (of real dimension $>2$). See Farrell's "Lectures on Surgical Methods in Rigidity".
2. Real Pontryagin classes of complete hyperbolic (or more generally conformally flat) manifolds vanish.
3. Orientable closed hyperbolic manifolds have even Euler characteristic, while there is a complex hyperbolic surface constructed by Mumford of Euler characteristic 3. (Proof: every orientable closed manifold of dimension not divisible by 4 has even Euler characteristic, and in dimensions divisible by 4 
the Euler characteristic is the Betti number in the middle dimension, which vanishes for hyperbolic manifolds because their signature iz zero by the fact 2 above).


