This posts documents my attempt at understanding the paper of the same name written by my friend and his advisor. I thank Frieder for giving me an overview of what was going on and the post will broadly follow his explanation.
When students learn about differential geometry of curves and surfaces, one of the things taught is that parallel transport along a loop may not return the same initial vector, and on the sphere this is the result of positive curvature. In higher dimensions, one can write down the curvature tensor (of a Riemannian manifold with a Levi-Civita connection)
\[ R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w.\]
The curvature tensor is a linear transformation of the tangent space and measures the failure of covariant derivatives to commute. If this seems rather abstract to you, you may be comforted to know it does to a lot of other people including me. Here's a more geometric approach: fix a point $w$ and two vectors in the tangent space $T_w$ and consider the image of a ball in the plane $\sigma$ they generate under the exponential map. This is a subsurface for a sufficiently small ball, and one can compute (Gaussian) curvature at $w$. If $u,v$ are two linearly independent vectors in $\sigma$ then
\[ K(\sigma )=K(u,v)/|u\wedge v|^{2}{\text{ where }}K(u,v)=\langle R(u,v)v,u\rangle \]
and one can check that the curvature tensor can be completely recovered from this data.
A popular and successful approach to studying the topology of manifolds is studying the sort of metrics that they admit. In particular, one can ask whether there is a distinguished or best metric on a manifold, where distinguished could mean anything from easiest to understand to a more technical thing like minimises some functional on the space of metrics. Usually one interprets this to mean constant curvature. However, manifolds with constant sectional curvature are in a sense classified: their universal covers are isometric to one of the sphere, Euclidean space, or hyperbolic space (in the appropriate dimension).
This is, however, not where the story ends. In dimension 2 there isn't very much room for things to go wrong, but in higher dimensions there are different notions of curvature which end up agreeing in dimension 2. We will discuss a couple more.
One can consider scalar curvature, which is defined to be the trace of the Ricci tensor. This sees the volume of balls: when the scalar curvature is positive at a point, the volume of a small geodesic ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is larger than it would be in Euclidean space. However, it sees very little of the Ricci tensor, and it turns out very little can be said in general about manifolds that admit metrics of constant scalar curvature. See the following two theorems:
Theorem (Kazdan-Warner): Given a smooth closed manifold $M$ of dimension at least 3, exactly one of the following holds (all functions are maps $f: M \to mathbb{R}$):
- Every function on $M$ is the scalar curvature of some metric on $M$.
- A function on $M$ is the scalar curvature of some metric on $M$ if and only if it is either identically zero or negative somewhere.
- A function on $M$ is the scalar curvature of some metric on $M$ if and only if it is negative somewhere.
Theorem (Yamabe Problem): Let $(M,g)$ be a closed smooth Riemannian manifold. Then there exists a positive and smooth function $f$ on $M$ such that the Riemannian metric $fg$ has constant scalar curvature.
Much more can be said about manifolds admitting metrics of constant positive scalar curvature and this is a huge area of research. For example, Perelman used the Ricci flow with surgery to show that 3-manifolds admitting metrics of constant positive scalar curvature are in fact homeomorphic to the 3-sphere. We will, however, leave this for another time.
The third notion that we will discuss today is Ricci curvature. In dimension at least 4 this also doesn't completely describe the Ricci tensor, but plenty can be said here. Given an orthonormal basis $\{e_{i}\}$ in the tangent space at $p$ we define the Ricci curvature
\[\mathrm {Ric} (u)=-\sum _{i}R(u,e_{i})e_{i}.\]
Ricci curvature is a linear operator on tangent space at a point which does not depend on the choice of orthonormal basis. Metrics of constant Ricci curvature are also known as Einstein metrics. I'll leave the question of why we care about these to
MSE, but it seems that these are a good compromise between sectional and scalar curvature.
The paper gives an explicit construction of infinitely many manifolds in every dimension at least 4 which admit negatively curved Einstein metrics but do not admit a hyperbolic (ie constant sectional curvature) metric. Previously this was done in dimension 4 by Fine-Premoselli using extremely involved analytic methods, and Gromov-Thurston proved the existence of infinitely many such manifolds in every dimension, but this wasn't constructive. This is naturally broken up into two parts: the existence of the Einstein metrics and the non-existence of hyperbolic metrics.
Being an Einstein metric is equivalent to being a zero of a certain linear operator, which is called the Einstein operator. Fine and Premoselli perturb the hyperbolic metric to get a metric on the branched cover which 'almost' satisfies the relevant equations and then appeal to a sort of inverse function theorem to show that the metric on the branched cover can be perturbed to give an Einstein metric. The exact construction is as follows:
Consider $\mathbb{H}^n$ and fix a totally geodesic copy $S \subset \mathbb{H}^n$ of $\mathbb{H}^{n−2}$ . Let $g_S$ denote the hyperbolic metric of $S$. Then in exponential normal coordinates centered at $S$, the hyperbolic metric of $\mathbb{H}^n$ is given by $g_{\mathbb{H}^n} = dr^2 + \sinh^2(r)d \theta^2 + \cosh^2(r)g_S$. Using the change of variables $u = \cosh(r)$, this becomes $du^2+ (u^2 − 1)d\theta^2 + u ^2 g_S$, which is defined for $(u, \theta) \in (1,\infty) \times S^1$ . Fine–Premoselli consider metrics of the form $du^2+ V (u)d\theta^2 + u^2 g_S$ where $V$ is a positive smooth function. They show that this gives rise to an Einstein metric iff $V (u) = u^2 − 1 + \frac{a}{u^{n−3}}$ for some real $a$. The approximate Einstein metric on cover of $M$ branched along $\Sigma$ is then obtained by interpolating between such a metric (defined on a tubular neighbourhood of $\Sigma$ by taking the image of perturbed metric on hyperbolic space) and the hyperbolic metric (defined away from $\Sigma$).
The reason Fine-Premoselli only do this is in dimension 4 is that to apply the relevant inverse function theorem they need to have small $L^2$ norm of $\mathrm{Ric}(g)+(n-1)g$, which they can only do in dimension 4. The innovation in this paper is to use subgroup separability properties of arithmetic groups to keep the $L^2$ norm small as the branching degree grows large. Recall that a subgroup $H \leq G$ is said to be separable in $G$ if for any $g \in G \setminus H$ there is a finite index subgroup $K$ of $G$ such that $g \in K$ and $H \leq K$. This is equivalent to $H$ being closed in the profinite topology of $G$.
Hamenstadt-Jackel show the following:
Proposition For each $n \geq 4$ and any standard arithmetic hyperbolic manifold $M$ , there is a sequence of finite covers $(M_k ) _{k \in \mathbb{N}}$ of $M$ containing closed embedded totally geodesic submanifolds $\Sigma_k \subset H_k \subset M_k$ with the following properties:
- The manifolds $\Sigma_k$ are all isometric, and they are of codimension 2.
- $\Sigma_k$ is null-homologous in the embedded connected hypersurface $H_k \subset M_k$;
- $\Sigma_k$ has at most two connected components;
- We have $\lim_{k \to \infty} \frac{diam(\Sigma_k)}{R_k^{\nu}}= 0$, where $R_k^{\nu}$ is the normal injectivity radius of $\Sigma_k \subset M_k$, and $diam(\Sigma_k)$ is the maximum of the diameters of the connected components of $\Sigma_k$.
Subgroup separability is used as follows: by Bergeron-Haglund-Wise, after passing to a finite cover if necessary we may find a hyperplane in the universal cover whose image in the finite cover is a standard arithmetic hyperbolic manifold which is a retract of the fundamental group of the finite cover. It is a fact that in any hyperbolic group and for any real $R$, there are only finitely many conjugacy classes of elements of translation length at most $R$. Something similar is true here: there are only finitely many elements which cause the normal injectivity radius of $\Sigma_k$ to be small, so subgroup separability allows one to pass to a finite cover one and get rid of this while preserving the codimension 2 submanifold. One then applies this to show that the corresponding $L^2$ norm is sufficiently small.
Now all that is left to show is that the corresponding branched cover $X$ can't be homeomorphic to any locally symmetric manifold. We first show that if it is homeomorphic to one, it must be a hyperbolic manifold. Since the fundamental group is Gromov hyperbolic, if it is homeomorphic to a locally symmetric manifold it must be a rank 1 manifold (otherwise there would be $\mathbb{Z}^2$ subgroups.
It remains to show that X is not homeomorphic to any complex-, quaternionic- or Cayley-hyperbolic manifold. If X is homotopy equivalent to a complex hyperbolic manifold N , then there is a degree $d \geq 2$ map $\pi ∶ N \to M$ . Since $M$ has constant negative curvature, the map $\Pi$ is homotopic to a harmonic map. But by a theorem of Sampson, any harmonic map from a compact Kähler manifold into a real hyperbolic manifold is trivial in homology of dimension larger than two, which contradicts the fact that the degree of the map $\Pi$ is positive (unless $n = 2$ and $N$ is also real hyperbolic).
Novikov showed that the rational Pontryagin classes, and hence the Pontryagin numbers, are a homeomorphism invariant. By a result of Lafont–Roy all Pontryagin numbers of X vanish, while it is a consequence of the Hirzebruch proportionality principle that closed quaternionic- or Cayley-hyperbolic manifolds have some non-zero Pontryagin numbers. Therefore, $X$ can also not be homeomorphic to a quaternionic- or Cayley-hyperbolic manifold.
The authors go on to show
Theorem Let $M$ be an oriented closed hyperbolic manifold of dimension $n \geq 4$ and let $H \subset M$ be a totally geodesic embedded hyperplane. Assume that $H$ is contained in the fixed point set of an orientation reversing isometric involution $\iota$ and that $H$ contains a (possibly disconnected) embedded totally geodesic hyperplane $\Sigma$ which is homologous to zero in $H$. Then for at most one $d \in 4\mathbb{Z}$, the cyclic $d$-fold covering of $M$ branched along $\Sigma$ can be homeomorphic to a hyperbolic manifold.
The strategy is that if there were different degrees $d_1 \neq d_2$of branched covers that were hyperbolic, one could glue them up to produce another hyperbolic manifold. Furthermore, by analysing the action of the cyclic group describing the branching one can compute that the totally geodesic codimension 2 submanifolds of the pieces intersect at an angle of either $\frac{2\pi}{d_1}$ or $\frac{2\pi}{d_2}$. Separately, Seifert-van Kampen implies there are outer automorphisms of the fundamental group of order 2 and $\frac{d_1+d_2}{2}$, which must come from isometries of the hyperbolic manifold by Mostow Rigidity. This then forces all the intersection angles to be the same, which is a contradiction.
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