# Are negatively pinched manifold locally conformally flat?

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?

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 $O(3,1)$.

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

1. 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".

2. 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 $\mathbb Z$.).

• On the other hand, some (maybe all?) of Gromov-Thurston examples admit conformally-flat metrics. Thus, it could be that Pontryagin classes are the only obstructions to existence of conformally-flat metrics on closed negatively curved manifolds which are sufficiently pinched. – Misha Sep 21 '12 at 4:41
• @Igor, Is it true that "locally" conformally flat is enough for vanishing Pontryagin classes? I only know Pontryagin classes differs by exact forms for globally conformally metrics. – J. GE Sep 21 '12 at 9:02
• GB: here is the way I recall it. By Chern-Weil theory one can write Pontryagin forms (which represent the Pontryagin classes) in terms of components of the curvature tensor, and it turns out that if Weyl tensor vanishes, then so does the Pontryagin forms. I do not have a reference handy. This is a local computation, so as long as a manifold is locally conformally flat, the Pontryagin form vanishes. – Igor Belegradek Sep 21 '12 at 10:17
• @Misha: It is an interesting question whether there is a conformally flat version of Ontaneda's result. Somehow I am sceptical that Pontryagin classes is the only obstruction. – Igor Belegradek Sep 21 '12 at 10:20
• @Igor, For locally conformally flat metric, one can choose a flat connection in a nbhd $U_x$ of each point $p\in X$ such that the Pontrjagin form vanishes at $U_x$, I don't see why the Pontryagin class vanishes globally. ps. I read Chern-Simons "Characteristic Forms and Geometric Invariants" Ann. 1974. On Theorem 4.5 they proved the conformal invariance for Pontrjagin class for globally conformal change of metric. Did I miss something? – J. GE Sep 21 '12 at 19:45

Another easy locally homogeneous counterexample to the original question is given by complex hyperbolic manifolds. this includes compact examples. Complex hyperbolic manifolds are Einstein. Curvature tensor of any manifold decomposes into its Weyl part+Ricci part +scalar part. Thus, a conformally flat Einstein manifold must necessarily have scalar curvature operator and hence have constant sectional curvature in dimensions above 2. This is definitely not the case for complex hyperbolic manifolds so they are not locally conformally flat. Also, as Igor mentioned Chern-Weil theory in dimension 4 says that $sig(M^4)=\frac{1}{12\pi^2}\int_M(|W^+|^2-|W^-|^2)$, where $W^\pm$ are self-dual and anti-self-dual parts of $W$. Complex hyperbolic manifolds are conformally semi-flat (i.e they have $W^-=0$) which can be easily derived from the fact that their curvature tensors are $U(2)$ invariant. thus, for a closed complex hyperbolic 4-manifold its signature is $sig(M^4)=\frac{1}{12\pi^2}\int_M(|W^+|^2\ne 0$. moreover, the integrant is just a constant (by homogeneity).