
Here we prove that under a lower sectional curvature bound, a sequence of Riemannian manifolds diffeomorphic to the standard m-dimensional torus cannot converge in the Gromov–Hausdorff sense to a closed interval.
The proof is done by contradiction by analyzing suitable covers of a contradicting sequence, obtained from the Burago–Gromov–Perelman generalization of the Yamaguchi fibration theorem.
Citation: Sergio Zamora. Tori can't collapse to an interval[J]. Electronic Research Archive, 2021, 29(4): 2637-2644. doi: 10.3934/era.2021005
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Here we prove that under a lower sectional curvature bound, a sequence of Riemannian manifolds diffeomorphic to the standard m-dimensional torus cannot converge in the Gromov–Hausdorff sense to a closed interval.
The proof is done by contradiction by analyzing suitable covers of a contradicting sequence, obtained from the Burago–Gromov–Perelman generalization of the Yamaguchi fibration theorem.
The study of Riemannian manifolds with sectional curvature bounded below naturally leads to the study of Alexandrov spaces in part due to the following well known results.
Theorem 1.1. ([9], Theorem 5.3). Let
Theorem 1.2. ([4], Section 8). Let
In this situation, results by Perelman and Yamaguchi show that in many cases, the topology of the limit is closely tied to the topology of the sequence.
Theorem 1.3. [15], [11]. Let
Theorem 1.4. ([16], Main Theorem). Let
Even with these two powerful theorems, collapsing under a lower curvature bound is still far from being well understood, specially when the limit space has singularities or boundary. At the local level, Vitali Kapovitch has successfully studied the behaviour of collapse.
Theorem 1.5. [12]. Let
At the global level, Mikhail Katz recently proved that the
Theorem 1.6. [14]. Let
The goal of this note is to prove the following generalization of Theorem 1.6.
Theorem 1.7. Let
Remark 1. Let
Theorem 1.7 represents a step towards the following conjecture. Theorem 1.3 implies the case
Conjecture 1. Let
Remark 2. Gromov and Lawson showed in [10] that if a Riemannian manifold diffeomorphic to the
The structure of this note is as follows: In section 2, we give a consequence of our main theorem. In section 3 we give the necessary definitions and preliminaries. In section 4 we give the proof of Theorem 1.7.
A little bit more can be said about manifolds admitting flat metrics. Recall Bieberbach Theorem and the definition of holonomy group. An elegant proof can be found in [6].
Theorem 2.1. Let
0→Zm→π1(M)→HM→0. |
The group
Theorem 2.2. Let
For the proof of Theorem 2.2, we will need the following elementary result.
Lemma 2.3. ([8], Section 6) Let
Proof of Theorem 2.2. Let
Theorem 2.2 implies in particular that if the holonomy group
Theorem 2.4. ([1], Theorem 3). Let
Gromov–Hausdorff distance was introduced to quantitatively measure how far two metric spaces are from being isometric.
Definition 3.1. Let
dGH(A,B):=infφ,ψdH(φ(A),ψ(B)), |
where
We refer to ([3], Chapter 7) for the basic theory about Gromov–Hausdorff distance, including the following equivalence of convergence with respect to such metric.
Theorem 3.2. Let
limn→∞supx,y∈Xn|d(fn(x),fn(y))−d(x,y)|=0, |
and
limn→∞supx∈Xinfy∈Xnd(x,fn(y))=0. |
A sequence of functions
In a compact Alexandrov space
d(γ1,γ2):=lims,t→0+cos−1(d(γ1(s),γ2(t))2−s2−t2st)∈[0,π]. |
The set of all minimizing geodesics in
For
Theorem 3.3. ([4], Section 9). For small enough
Since the definition of an almost non-negatively curved manifold in the generalized sense is technical and we will not use it, we will omit it. The only result we will need regarding such manifolds is the following.
Theorem 3.4. ([16], Pinching Theorem). The first Betti number of an
We will use two elementary facts about comparison geometry. The first one is called the four point Alexandrov condition.
Lemma 3.5. ([4], Section 2). Let
θ1+θ2+θ3≤2π. |
This condition is called the Alexandrov condition for the quadruple
Another ingredient is the Bishop–Gromov inequality.
Theorem 3.6. ([2], p. 253). Let
Vol(BR(p))Vol(Br(p))≤Vol(BR(q))Vol(Br(q)). |
Lemma 3.7. ([9], Proposition 5.28) Let
We will also use a simple version of Gromov's systolic inequality.
Theorem 3.8. ([7], Section 1). Let
Assume by contradiction, that there is a sequence
Lemma 4.1. For any
Proving Theorem 1.7 for
The sequence
Let
By Theorem 3.6, applied to
C3≤Vol(B3L(˜qn))Vol(BL−3ε(˜qn))≤Vol(B3L(q))Vol(BL−3ε(q)). |
The right hand side only depends on
Fix a small
Let
First assume that there are no loops of Type Ⅰ. This would mean that the inclusion
j:f−1n([0,L−ε])→Xn |
induces a map at the level of fundamental groups such that
j∗(π1(f−1n([0,L−ε])))⊂i∗(π1(Fn)). |
This implies that the preimage of
Now assume that there are two loops
d(˜pn,an)≈d(˜pn,bn)≈d(an,bn)≈L |
d(qn,˜pn)≈d(qn,an)≈d(qn,bn)≈L/2, |
where the error in the above approximations is of the order of
With this, we see that in
The author would like to thank Raquel Perales and Anton Petrunin for stimulating his interest in Conjecture 1, and an anonymous reviewer whose comments improved the exposition of this paper.
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