Are negatively pinched manifold locally conformally flat? - MathOverflow

Are negatively pinched manifold locally conformally flat?

2012-09-20T21:47:52Z

One knows that hyperbolic manifolds are locally conformally flat. How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy: $$ -\Lambda \le K \le -\lambda$$ for $\Lambda>\lambda$.

How about sufficiently pinched, i.e. $\Lambda/\lambda=1+\epsilon$ for $\epsilon$ small?

Are they have vanished Pontryagin classes?

Answer by Ian Agol for Are negatively pinched manifold locally conformally flat?

2012-09-20T22:32:02Z

No in dimensions $\geq 3$. To be conformally flat in 3 dimensions, the Cotton tensor must to vanish, and in dimensions $\geq 4$, the Weyl tensor must vanish.

Maybe the point of your question though is to ask why is the Cotton or Weyl tensor non-vanishing? I don't have a good explanation for this.

Here's a special example in 3 dimensions. Consider the metric $$dr^2+e^{2k_1r}dx_1^2+e^{2k_2r}dx_2^2.$$ This metric is homogeneous, and when $k_1>0, k_2>0, k_1\neq k_2$, the metric is not conformally flat. The 3 principal sectional curvatures are $-k_1^2, -k_2^2, -k_1k_2$, and the isometry group is a solvable group. If the metric were conformally flat, then this solvable group would embed into $O(3,1)$ by Liouville's theorem. However, one can check that this solvable group does not embed by analyzing the Lie algebra and comparing it to the Lie algebras of the solvable subgroups of $O(3,1)$.

Answer by Igor Belegradek for Are negatively pinched manifold locally conformally flat?

2012-09-21T00:52:29Z

Regarding vanishing rational Pontryagin classes (which do vanish for conformally flat manifolds):

Recent result of Ontaneda gives examples in each dimension $\ge 4$ of closed manifolds with nonzero rational Pontryagin classes that are pinched arbitrary close to $-1$, see Corollary 4 of his paper "Pinched smooth hyperbolization".
On the other hand, if you restrict topology of your negatively pinched $n$-manifolds in a suitable way, then one can prove vanishing of Pontryagin classes for pinching close enough to $-1$. For example, for closed manifolds of uniformly bounded simplicial volume, if the pinching is close enough to $-1$, the manifold is diffeomorphic to a hyperbolic one (this is due to Gromov), and hence has zero Pontryagin classes. Long ago I proved similar results in the noncompact case, e.g. if you fix the fundamental group, and the dimension, and assume the metric is complete and the fundamental group is hyperbolic, then the Pontryagin classes vanish for pinching close to $-1$, see here. (I should mention that my proof depends on an accessibility result of Delzant-Potyagailo in which a gap was recently discovered by Louder-Touikan who announced a fix for hyperbolic groups, see here. Without the fix I can only handle the case of hyperbolic groups that do not split over {1} ot $\mathbb Z$.).