Negative sectional curvature and constant curvature

Question

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?

Answer 1

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).

Answer 2

To make things interesting, I will consider two classes of manifolds (one strictly larger than the other) without compactness assumption:

1. connected manifolds (dimension is finite but not fixed) admitting complete, negatively pinched Riemannian metrics

2. connected manifolds (dimension is finite but not fixed) admitting metrics of constant negative curvature.

In this setting there are exactly 3 known (to me) distinctions between fundamental groups $\pi$ of manifolds which belong to these classes, the first is very simple and the other two are quite subtle:

In Class 2, we have:

a) Every nilpotent subgroup of $\pi$ is virtually abelian.

b) $\pi$ has Haagerupp property: It admits a proper isometric affine action on a Hilbert space.

c) If $K$ is a closed Kaehler manifold and $\rho: \pi_1(K)\to \pi$ is a homomorphism, then either $\rho$ factors through the fundamental group of a 1-dimensional complex orbifold or the image of $\rho$ is virtually abelian.

There are other distinctions, but they all could be reduced to these three.

All three properties are known to be violated in Class 1, even if one restricts to manifolds of finite volume. For instance, quaternionic-hyperbolic lattices (acting on ${\bf HH}^n, n\ge 2$) have property T, which is a strong negation of Haagerup. Counter-examples to (c) come from complex-hyperbolic manifolds. Thus, in all three cases, the distinction could be traced to locally-symmetric spaces of rank 1.

Conjecturally, there is one more nontrivial distinction:

d) Finitely generated groups $\pi$ in Class 2 are residually finite. (Same for the fundamental groups of other locally-symmetric spaces.)

It is expected that fundamental groups in Class 1 could violate this property, but this is a major open problem, even if one considers the class of hyperbolic groups, where things are more flexible.

There are other distinctions between classes 1 and 2, coming from topology of the manifold and not from the fundamental group, when you look, say, at manifolds which are open disk bundles with hyperbolic base or intersection pairing of open 4-manifolds. You can read this survey to find more about these.