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Tori can't collapse to an interval

  • Received: 01 October 2020 Published: 11 January 2021
  • Primary: 53C23, 53C20; Secondary: 53C21

  • 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 Xn be a sequence of closed m-dimensional Riemannian manifolds with sectional curvature c and diameter D. Then there is a subsequence that converges in the Gromov–Hausdorff sense to a compact space X.

    Theorem 1.2. ([4], Section 8). Let Xn be a sequence of closed m-dimensional Riemannian manifolds with sectional curvature c. If the sequence Xn converges in the Gromov–Hausdorff sense to a compact space X, then X is an -dimensional Alexandrov space of curvature c with m.

    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 Xn be a sequence of closed m-dimensional Riemannian manifolds with sectional curvature c. If the sequence Xn converges in the Gromov–Hausdorff sense to a compact m-dimensional Alexandrov space of curvature c, then there is a sequence fn:XnX of Gromov–Hausdorff approximations such that fn is a homeomorphism for large enough n.

    Theorem 1.4. ([16], Main Theorem). Let Xn be a sequence of closed m-dimensional Riemannian manifolds with sectional curvature c. If the sequence converges in the Gromov–Hausdorff sense to a closed Riemannian manifold X, then there is a sequence of Gromov–Hausdorff approximations fn:XnX such that fn is a locally trivial fibration for large enough n.

    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 Xn be a sequence of closed m-dimensional Riemannian manifolds with sectional curvature c that converges in the Gromov–Hausdorff sense to a compact Alexandrov space X of curvature c. Then for any x0X, any sequence of Gromov–Hausdorff approximations fn:XnX, and any sequence of points xnf1n(x0), there is an r0(x0)>0 such that for large enough n, the closed ball Br0(xn) is a manifold with boundary, simply homotopic to a finite CW-complex of dimension m.

    At the global level, Mikhail Katz recently proved that the 2-dimensional torus cannot collapse to a segment.

    Theorem 1.6. [14]. Let gn be a sequence of Riemannian metrics of sectional curvature 1 in the 2-dimensional torus M. Then it cannot happen that the sequence (M,gn) converges in the Gromov–Hausdorff sense to an interval [0,L].

    The goal of this note is to prove the following generalization of Theorem 1.6.

    Theorem 1.7. Let gn be a sequence of Riemannian metrics of sectional curvature 1 in the m-dimensional torus M. Then it cannot happen that the sequence (M,gn) converges in the Gromov–Hausdorff sense to an interval [0,L].

    Remark 1. Let Φn be the group of isometries of C generated by zz+2i and z¯z+1n. The quotient Wn=C/Φn is a flat Klein bottle and the sequence Wn converges in the Gromov–Hausdorff sense to [0,1] (see Figure 1), so Theorem 1.7 is false if one replaces the m-dimensional torus by the Klein bottle.

    Figure 1.  Flat Klein bottles can converge to an interval.

    Theorem 1.7 represents a step towards the following conjecture. Theorem 1.3 implies the case =m, and Theorem 1.7 the case =1.

    Conjecture 1. Let gn be a sequence of Riemannian metrics of sectional curvature 1 in the m-dimensional torus M such that the sequence (M,gn) converges to a compact -dimensional Alexandrov space X. Then X is homeomorphic to an -dimensional torus.

    Remark 2. Gromov and Lawson showed in [10] that if a Riemannian manifold diffeomorphic to the m-dimensional torus has scalar curvature 0, then it is flat. The Mahler compactness Theorem asserts that if gn is a sequence of flat Riemannian metrics in the m-dimensional torus M, such that the sequence (M,gn) converges in the Gromov–Hausdorff sense to a compact space X, then X is a flat torus (see [5], p.137). Therefore Conjecture 1 is known to be true if we replace sectional curvature 1 by scalar curvature 0.

    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 M be a flat closed m-dimensional manifold. Then its fundamental group fits in an exact sequence

    0Zmπ1(M)HM0.

    The group Zm is the only maximal abelian normal subgroup of π1(M). The group HM is finite and it is called the holonomy group of M. The cover associated to Zmπ1(M) is a flat torus.

    Theorem 2.2. Let M be a closed m-dimensional manifold that admits a flat metric. If there is a sequence gn of Riemannian metrics with sec(M,gn)1 such that (M,gn) converges to an interval [0,L], then the holonomy group HM has a subgroup of index 2.

    For the proof of Theorem 2.2, we will need the following elementary result.

    Lemma 2.3. ([8], Section 6) Let Γ be a finite group and Xn a sequence of compact metric spaces converging in the Gromov–Hausdorff sense to the space X. Assume we have a sequence of isometric group actions ΓIso(Xn). Then there is an isometric group action ΓIso(X) such that a subsequence of Xn/Γ converges in the Gromov–Hausdorff sense to X/Γ.

    Proof of Theorem 2.2. Let Xn be the torus metric cover of (M,gn) with Xn/HM=(M,gn). The diameter of Xn is bounded above by 2L|HM| for large enough n, hence by Theorems 1.1 and 1.2, Xn converges up to subsequence to an Alexandrov space X of dimension m. By Lemma 2.3, there is an isometric group action HMIso(X) such that X/HM=[0,L]. The quotient of a compact finite dimensional Alexandrov space by a finite group is another compact Alexandrov space with the same dimension. This means that X is a compact 1-dimensional Alexandrov space, so it is a circle or a closed interval. By Theorem 1.7, X is a circle, and the action HMIso(X) is either cyclic or dyhedral. The quotient of X by a dihedral group is an interval, and the quotient of X by a cyclic group is a shorter circle. Since X/HM=[0,L], the image of HM in Iso(X) is a dihedral group, which has a subgroup of index 2.

    Theorem 2.2 implies in particular that if the holonomy group HM is simple, or has odd order, then M cannot collapse to an interval under a lower sectional curvature bound. The following Theorem by Auslander and Kuranishi tells us the relevance of Theorem 2.2.

    Theorem 2.4. ([1], Theorem 3). Let H be a finite group. Then there is a flat manifold M with HM=H.

    Gromov–Hausdorff distance was introduced to quantitatively measure how far two metric spaces are from being isometric.

    Definition 3.1. Let A,B be two metric spaces. The Gromov–Hausdorff distance dGH(A,B) between A and B is defined as

    dGH(A,B):=infφ,ψdH(φ(A),ψ(B)),

    where dH denotes the Hausdorff distance, and the infimum is taken over all isometric embeddings φ:AC, ψ:BC into a common metric space C.

    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 Xn be a sequence of compact metric spaces. The sequence Xn converges in the Gromov–Hausdorff sense to a compact metric space X if and only if there is a sequence of maps fn:XnX such that

    limnsupx,yXn|d(fn(x),fn(y))d(x,y)|=0,

    and

    limnsupxXinfyXnd(x,fn(y))=0.

    A sequence of functions fn satisfying the above properties are called Gromov–Hausdorff approximations.

    In a compact Alexandrov space X of dimension , one can quantify how degenerate a point pX is by studying the Gromov–Hausdorff distance between its space of directions ΣpX and the standard sphere S1. To construct ΣpX, one needs to put a metric on the set of geodesics in X emanating from p. Given two minimizing geodesics γi:[0,δi]X parametrized by arc length with γi(0)=p for i=1,2, we define the angular distance between them as

    d(γ1,γ2):=lims,t0+cos1(d(γ1(s),γ2(t))2s2t2st)[0,π].

    The set of all minimizing geodesics in X starting from p equipped with the angular metric form a semi-metric space SpX. The space ΣpX is defined as the metric completion of the metric space asociated to SpX.

    For δ>0, we say that a point p is δ-regular if dGH(ΣpX,S1)<δ. The set of δ-regular points Uδ(X)X form an open dense set, and for small enough δ, they form an -dimensional (topological) manifold. Burago, Gromov, and Perelman noticed that Theorem 1.4 has a version for when X is singular.

    Theorem 3.3. ([4], Section 9). For small enough δ(m,c) the following holds. Let Xn be a sequence of closed m-dimensional Riemannian manifolds with sectional curvature c converging in the Gromov–Hausdorff sense to a compact space X. Then for any compact KUδ(X) there is a sequence of Gromov–Hausdorff approximations fn:XnX such that for large enough n, fn|f1n(K) is continuous and morover, it is a locally trivial fibration with fiber Fn, a compact almost nonnegatively curved manifold in the generalized sense of dimension m (ANNCGS(m)) (see [13], Definition 1.4.1).

    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 ANNCGS(d) is d.

    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 N be an m-dimensional Riemannian manifold with sectional curvature c and Mm(c) be the simply connected complete m-dimensional Riemannian manifold of constant curvature c. For distinct points p,a1,a2,a3N and i{1,2,3}, we set a4=a1 and call θi the angle at ˜p of a triangle ˜ai˜p˜ai+1 in Mm(c) with d(˜p,˜ai)=d(p,ai), d(˜p,˜ai+1)=d(p,ai+1), d(˜ai,˜ai+1)=d(ai,ai+1). Then

    θ1+θ2+θ32π.

    This condition is called the Alexandrov condition for the quadruple (p;a1,a2,a3).

    Another ingredient is the Bishop–Gromov inequality.

    Theorem 3.6. ([2], p. 253). Let Z be an m-dimensional Alexandrov space of curvature c, and Mm(c) be the simply connected complete m-dimensional Riemannian manifold of constant curvature c. Then for 0<r<R, pZ, qMm(c), we have

    Vol(BR(p))Vol(Br(p))Vol(BR(q))Vol(Br(q)).

    Lemma 3.7. ([9], Proposition 5.28) Let Z be a compact semilocally simply connected length space, z0Z, η>0, and r=supzZd(z,z0). Then π1(Z,z0) is generated by the loops of length 2r+η.

    We will also use a simple version of Gromov's systolic inequality.

    Theorem 3.8. ([7], Section 1). Let N be a smooth closed aspherical manifold, and gn a sequence of Riemannian metrics on N such that the volumes of the spaces Zn=(N,gn) go to 0 as n. Then there is a sequence of noncontractible loops γn:S1Zn with lengths going to 0 as n

    Assume by contradiction, that there is a sequence Xn=(M,gn) as in Theorem 1.7 converging to an interval [0,L]. Applied to the limit space [0,L], Theorem 3.3 takes the following form.

    Lemma 4.1. For any ε>0, and large enough n(ε), there are continuous Gromov–Hausdorff approximations fn:Xn[0,L] such that f1n([ε,Lε]) is homeomorphic to the product [ε,Lε]×Fn, with Fn an ANNCGS(m1), and fn|f1n([ε,Lε]) being the projection onto the first factor.

    Proving Theorem 1.7 for m=2 is easier and gives an idea on how to get the general case. Fix a small ε (say, ε=L/100) and use Lemma 4.1. We see that the fibers Fn are homeomorphic to S1 (the only compact 1-dimensional manifold), exhibiting Xn as the connected sum of two surfaces S1#S2 (see Figure 2). Since the 2-dimensional torus is undecomposable, one of the surfaces, say S1, is homeomorphic to S2. This would imply that Yn:=f1n([0,Lε]) is homeomorphic to a disk, meaning that the inclusion YnXn is trivial at the level of fundamental groups. Therefore, when we take the universal covering ˜XnXn, the preimage of Yn consists of disjoint copies of Yn (one for each element of π1(Xn)=Z2).

    Figure 2.  The Fibration Theorem gives us a decomposition Xn=S1#S2.

    The sequence Xn collapses to a lower dimensional object, so the volume of Xn goes to 0 as n. Since the torus is aspherical, by Theorem 3.8 for any CN, and large enough n(C), there are non contractible loops γn:S1Xn of length L/C. Since Yn is contractible and γn is noncontractible, we have γn(S1)Yn. Let xn=γn(1), and ˜xn one of its preimages in ˜Xn. Since Z2=π1(Xn) has no torsion, there are at least C/3 elements of the orbit of ˜xn in the ball BL/2(˜xn).

    Let qnf1n([0,ε]), and ˜qn˜Xn its preimage closest to ˜xn. The ball BL3ε(˜qn) is isometric to BL3ε(qn). However, the ball B3L(˜qn) contains at least C/3 disjoint isometric copies of BL3ε(qn).

    By Theorem 3.6, applied to Z=˜Xn, c=1, p=˜qn, r=L3ε, R=3L, we get for any qMm(1),

    C3Vol(B3L(˜qn))Vol(BL3ε(˜qn))Vol(B3L(q))Vol(BL3ε(q)).

    The right hand side only depends on m,L,ε, so the above inequality cannot hold if C is large enough, which is a contradiction.

    Fix a small ε (to be chosen later) and use Lemma 4.1. By Theorem 3.4 we have that for large n, the image of the morphism i:π1(Fn)π1(Xn) induced by the inclusion i:FnXn has corank at least 1. Let ˜Xn be the cover of Xn with Galois group Γn:=π1(Xn)/iπ1(Fn). Observe that by construction, the preimage of f1n([ε,Lε]) in ˜Xn consists of disjoint copies of itself.

    Let pn be a point in f1n(L/2) and ˜pn a lift in ˜Xn. Let S be the set of loops in Xn based at pn of length L+10ε. By Lemma 3.7, for large enough n, S generates π1(Xn,pn). The elements of S whose image is contained in f1n([ε,Lε]) are homotopic to elements of iπ1(Fn) and lift to loops in ˜Xn. Let S be the subset of S not homotopic to elements in iπ1(Fn). S generates Γn and consists of loops that go to one of f1n([0,ε]) or f1n([Lε,L]), but not both. We will call them Type Ⅰ or Type Ⅱ depending on whether they visit f1n([0,ε]) or f1n([Lε,L]).

    First assume that there are no loops of Type Ⅰ. This would mean that the inclusion

    j:f1n([0,Lε])Xn

    induces a map at the level of fundamental groups such that

    j(π1(f1n([0,Lε])))i(π1(Fn)).

    This implies that the preimage of f1n([0,Lε]) in ˜Xn consists of infinitely many disjoint copies of itself. Also, since Γn is abelian of positive rank, any set of generators contains an element of infinite order. Then there is a loop of Type Ⅱ of infinite order and we can conclude identically as in the 2-dimensional case.

    Now assume that there are two loops α, β of Type Ⅰ not equivalent in Γn. This means that they lift as paths ˜α, ˜β in ˜Xn with startpoint ˜pn, but distinct endpoints an, bn, respectively. Letting qn be an approximate midpoint of an and ˜pn in the image of ˜α we see that

    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 ε. This violates the Alexandrov condition for the quadruple (qn;˜pn,an,bn) if ε(L) was chosen small enough (see Figure 3).

    Figure 3.  The configuration (qn;˜pn,an,bn) violates the Alexandrov condition.

    With this, we see that in S there is exactly one loop of Type Ⅰ and one loop of Type Ⅱ modulo iπ1(Fn). Observe that the inverse in Γn of the loop of Type Ⅰ is also a loop of Type Ⅰ, but there is only one loop of Type Ⅰ in Γn, so it is its own inverse, same for the loop of Type Ⅱ. But Γn is abelian of positive rank, so it cannot be generated by two elements of order 2.

    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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