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## Collapsing three-manifolds under a lower curvature bound (2000)

Venue: | J. DIFFERENTIAL GEOM |

Citations: | 30 - 3 self |

### Citations

501 |
The geometries of 3-manifolds
- Scott
- 1983
(Show Context)
Citation Context ... ∞ and that Mi does not admit a geometric structure. We may assume that each Mi is orientable. Passing to a subsequence, we may assume that Mi collapses to a compact Alexandrov space X. If X is a point, it follows from [13] together with the infiniteness assumption on the fundamental groups that Mi admits a geometric structure modelled on either S1×R, R3 or Nil, a contradiction. If dimX = 1 or 2, then Theorems 0.2, 28 takashi shioya & takao yamaguchi 0.3 and 0.5 together imply that Mi is homeomorphic to a Seifert fibred space or a infrasolvmanifold, and hence admits a geometric structure (see [29]). q.e.d. In view of the above proof, the infiniteness assumption on the fundamental groups in Corollary 0.9 can be replaced by a lower diameter bound. Namely we have the following corollary by the same argument. Corollary 6.2. For any positive numbers δ ≤ D, there exists a constant " = "(δ, D) > 0 such that if a closed, prime three-manifold admits a Riemannian metric contained in M(3, D) with diameter ≥ δ and volume < ", then it admits a geometric structure modelled on one of the seven geometries except H3. Proof of Corollary 0.10. This is done by contradiction. Suppose that the corollary doe... |

304 |
Structures Métriques Pour les Variétés Riemanniennes
- Gromov
- 1981
(Show Context)
Citation Context ...r is to completely characterize the topology of threedimensional Riemannian manifolds with a uniform lower bound of sectional curvature which converges to a metric space of lower dimension. 0. Introduction We study the topology of three-dimensional Riemannian manifolds with a uniform lower bound of sectional curvature converging to a metric space of lower dimension. Given a positive integer n and D > 0, let us consider the setM(n, D) of isometry classes of n-dimensional closed Riemannian manifolds M with sectional curvature K ≥ −1 and diameter diam(M) ≤ D. By the Gromov Precompactness Theorem [16], the closure of M(n, D) is compact with respect to the Gromov-Hausdorff distance. Thus any sequence Mi, i = 1, 2, . . . , in M(n, D) has a convergent subsequence whose limit is a compact Alexandrov space X of dimension ≤ n and curvature ≥ −1. We now assume that Mi itself converges to X, i is sufficiently large, and consider the following: Problem 0.1. Describe the topological structure of Mi by using the geometry and topology of X. Received October 22, 1999. 1991 Subject Classification. Primary 53C20, 53C23; Secondary 57N10, 57M99. Key words and phrases. the Gromov-Hausdorff convergence, Alex... |

289 |
The Geometry and Topology of 3-Manifolds
- Thurston
- 1978
(Show Context)
Citation Context ...there are eight geometric structures of three-manifolds modelled on S3, R3, H3, S2×R1, H2×R, SL2(R), Nil and Sol. Corollary 0.9. For any D > 0, there exists a constant " = "(D) > 0 such that if a closed, prime three-manifold with infinite fundamental group admits a Riemannian metric contained in M(3, D) with volume < ", then it admits a geometric structure modelled on one of the seven geometries except H3. collapsing three-manifolds 5 Observe that a hyperbolic manifold M does not collapse under K ≥ −1 because of the non-vanishing property ‖M‖ = 0 for simplicial volume (Gromov [17], Thurston [34]). Combining our results above, we obtain the following corollary on the existence of geometric structures for the elements of M(3, D). Corollary 0.10. All elements but finitely many homeomorphism classes in M(3, D) admit geometric structures. In the proofs of our results, we essentially use a critical pointrescaling argument to understand the topology of a small neighborhood of Mi converging to a small neighborhood of a singular point of X. When dimX = 2, as the limit space of the rescaled Mi, we have a three-dimensional complete open nonnegatively curved Alexandrov space Y which is a topolog... |

264 |
Spaces of constant curvature
- Wolf
- 2011
(Show Context)
Citation Context ...he new metric has diameter = 1. Thus we may consider that dimX = 1 or 2. Theorems 0.2, 0.3 and 0.5 then imply that Mi is homeomorphic to a Seifert fibred space or a infrasolvmanifold, and hence admits a geometric structure. q.e.d. 7. Construction of collapsing metrics In this section, we prove Theorem 0.6 by constructing collapsing metrics together with some examples. First we show that an infinite sequence of pairwise non-homeomorphic prism manifolds collapses to a closed interval under K = 1. Example 7.1. Let M be a prism manifold S3/Γ, where Γ ⊂ SO(4) is one of the following two types (see [36]): Type 1) Γ is generated by γ1 = ( R(1/m) 0 0 R(r/m) ) , γ2 = ( 0 I R(2/n) 0 ) , collapsing three-manifolds 29 where n is even, (n(r − 1), m) = 1, r ≡ r2 ≡ 1 mod m, (, n/2) = 1 and R(θ) = ( cos 2πθ − sin 2πθ sin 2πθ cos 2πθ ) . Type 2) Γ is generated by σ1 = ( R(u+vuv ) 0 0 R(v−uuv ) ) , σ2 = ( 0 I −I 0 ) , where v is even, (u, v) = 1. For Type 1), take distinct prime numbers p and q and consider the group Γpq of Type 1) defined by m = p, r = p − 1, n = 2q and = 1. The group generated by γ1 and the group generated by γ22 converge to distinct circle groups. This implies that the subgroup ... |

252 |
Ebin D. G.,: Comparison Theorems in Riemannian Geometry
- Cheeger
- 1975
(Show Context)
Citation Context ... Proof of Theorem 9.8 (1). Suppose that dimS = 1 and let Λ be the deck transformation group of the universal covering π : X → X. Then S is isometric to a circle and Λ Z. It is easy to see that π−1(S) is totally convex set isometric to R. The splitting theorem then implies that X is isometric to a product R×P . Let p1 : Λ → Isom(R), p2 : Λ → Isom(P ) be the projections. Then p1(Λ) acts on R by translation. If π−1(S) corresponds to R× {p}, then the point p is a fixed point of p2(Γ). This completes the proof of Theorem 9.8(1). q.e.d. In the Riemannian case, the Berger Comparison Theorem (cf. [7]) was used for the proof of Theorem 9.8(2). It is unknown if the Berger Comparison Theorem holds for Alexandrov spaces. Hence we need another argument for the proof. For the proof of Theorem 9.8(2), we consider the situation that C0 is of dimension n − 1. In what follows, we put C = C0 for simplicity. We say that a direction ξ ∈ Σp = Σp(X) at p ∈ C is normal to C if ∠(ξ, v) = π/2 for all v ∈ Σp(C). Lemma 9.9. Suppose that C = C0 has dimension n − 1. (1) Every point of C has at most two normal directions to C. (2) For a point p ∈ intC and a normal direction ξ ∈ Σp(X), there exists a locally iso... |

252 |
Four-manifolds with positive curvature operator
- Hamilton
(Show Context)
Citation Context ...0.5 improves the result of [13] in the case dimX = 0 previously stated in the following way. Corollary 0.7. Suppose that X is a point. Then a finite cover of Mi is homeomorphic to S1×S2, T 3, a nilmanifold or a simply connected Alexandrov space with nonnegative curvature. Conjecture 0.8. Any three-dimensional compact, simply connected, nonnegatively curved Alexandrov space without boundary which is a topological manifold is homeomorphic to a sphere. If Conjecture 0.8 is solved, everything will be clear about Problem 0.1 for n = 3. The above conjecture is certainly true in the Riemannian case ([20]). It is known by Thurston (cf. [35]) that there are eight geometric structures of three-manifolds modelled on S3, R3, H3, S2×R1, H2×R, SL2(R), Nil and Sol. Corollary 0.9. For any D > 0, there exists a constant " = "(D) > 0 such that if a closed, prime three-manifold with infinite fundamental group admits a Riemannian metric contained in M(3, D) with volume < ", then it admits a geometric structure modelled on one of the seven geometries except H3. collapsing three-manifolds 5 Observe that a hyperbolic manifold M does not collapse under K ≥ −1 because of the non-vanishing property ‖M‖ = 0 fo... |

250 | Manifolds of non-positive curvature
- Ballmann, Gromov, et al.
- 1985
(Show Context)
Citation Context ...ct and of nonnegative curvature, ( 1RY, y0) converges to the limit cone of Y as R → ∞. Therefore, for a µ > 1 and a sufficiently large R > 0, the rescaled annulus 1 Rδi A(pi;µ−1Rδi, µRδi) in Mi is dGH -close to the annulus A(o∞;µ−1, µ) in the limit cone for i large, where o∞ is the vertex of the limit cone. This together with a standard argument of critical point theory proves B(pi, Rδi) B(Si, Rδi). This completes the proof. q.e.d. Next we shortly discuss the ideal boundary of Y , where Y is as in Key Lemma 3.6. The ideal boundary Y (∞) of Y and the Tits metric ∠∞ on Y (∞) were defined in [2] (cf. [22], [32]). Let K be the asymptotic cone of Y defined as the pointed Gromov-Hausdorff limit of ("Y, y0) as " → 0 for a point y0 ∈ Y . Then K is the Euclidean cone over (Y (∞), ∠∞). collapsing three-manifolds 17 Lemma 3.7. There is an expanding map Σp → Y (∞). Proof. For a fixed " > 0 and for each x ∈ ∂B(p, "), take a minimal geodesic γi from pi to xi, where xi is a point in ∂B(pi, ") converging to x. Passing to a subsequence, we may assume that 1δi γi converges to a geodesic ray γx from y0 under the convergence ( 1δi B(pi, r), pi) → (Y, y0). Thus we have a map ϕ : ∂B(p, ") → Y (∞) d... |

242 |
Alexandrov spaces with curvature bounded from below
- Perelman, D
(Show Context)
Citation Context ...T 2. To investigate the topologies of Bi and Ci, we use the same discussion as (2) above. However, we only have 3 ≥ dimY ≥ dimX + 1 = 2 in this case. If dimY = 2, we apply the critical point-rescaling argument (2) to the convergence 1 δi B(pi, Rδi) → B(y0, R) with large R > 0 instead of Bi → B. This determines the topology of Bi and Ci and proves Theorem 0.5. collapsing three-manifolds 9 2. Preliminaries In this section, we present some results on Alexandrov spaces and the Gromov-Hausdorff convergence related with Alexandrov spaces, which will be needed in the subsequent sections. We refer to [4] for the basic materials and the details of the results on Alexandrov spaces mentioned below. First we give some basic definitions and notations. Let X be a geodesic space in the sense that every two points can be joined by a minimal geodesic. We assume that all geodesic have unit speed unless otherwise stated. For a fixed real number κ and a geodesic triangle ∆xyz in X with vertices x, y and z, we denote by ∆xyz a comparison triangle in the simply connected complete surface Mκ with constant curvature κ. This means that each side length of ∆xyz is equal to the corresponding one of ∆xyz. Here... |

185 |
On the structure of complete manifolds of nonnegative curvature
- Cheeger, Gromoll
(Show Context)
Citation Context ...t of X. When dimX = 2, as the limit space of the rescaled Mi, we have a three-dimensional complete open nonnegatively curved Alexandrov space Y which is a topological manifold. Here an Alexandrov space is called open if it is noncompact and without boundary. It is significant to determine the topology of such a space Y by using its soul S. Theorem 0.11. Let Y be a three-dimensional complete open Alexandrov space of nonnegative curvature. Suppose that Y is a topological manifold. Then Y is homeomorphic to the normal bundle N(S) of the soul S of Y . This extends the Cheeger-Gromoll Soul Theorem [9] in dimension three. Actually we classify all the three-dimensional complete open Alexandrov spaces with nonnegative curvature which are not necessarily topological manifolds (Theorem 9.6). This seems to be of independent interest. The organization of this paper is as follows: In Section 1, we sketch the essential idea of the proofs of Theorems 0.2, 0.3 and 0.5. In Section 2, we present some basic notions and results on Alexandrov spaces needed in the subsequent sections. The main body of this paper consists of two parts. In Part 1, we discuss the collapsing of three-dimensional Riemannian man... |

169 | The Geometry of Geodesics - Busemann - 1955 |

122 | Collapsing Riemannian manifolds while keeping their curvature bounded
- Cheeger, Gromov
(Show Context)
Citation Context ...rime number and v = p + 1. Then similarly one can verify that as p → ∞, S3(1)/Γp converges to [ 0, π/4 ] under K ≡ 1. The case dimX = 2 is covered by the following two examples. We denote by Dn(") and Sn−1(") = ∂Dn(") the n-disk and the (n−1)-sphere of radius ". Proof of Theorem 0.6. (I) Let Mn be a Seifert fibred space over an (n − 1)-dimensional smooth compact orbifold X without boundary. Then it admits a local S1-action, which defines a pure-polarized Fstructure on Mn. Therefore Mn admits a sequence of metrics which collapses to X with bounded curvature |K |≤ Λ for some constant Λ > 0. See [10] for details. (II) Let N3 be a Seifert fibred space over two-dimensional smooth compact orbifold X with boundary. We suppose that X has a product metric ∂X × [ 0, δ ) near the boundary ∂X. As in (I), one can construct a Riemannian metric h on N such that: (a) (N, h) collapses to X with |K |≤ Λ. (b) Near the boundary (N, h) is the product of a collar neighborhood of ∂X and S1("). 30 takashi shioya & takao yamaguchi Let D2 denote a disk with a metric such that: (c) The diameter of D2 is less than 10". (d) The curvature of D2 is nonnegative and its metric is a product metric S1(")× [ 0, δ )... |

85 |
A Generalized Sphere Theorem
- Grove, Shiohama
(Show Context)
Citation Context ...at πT ◦ fM gives a locally trivial fibre bundle on a neighborhood of p over a neighborhood of πT ◦ fM (p) in T , where πT denotes the nearest point projection to T . Since π : Tp → fX(Xδ) is homeomorphic on a small neighborhood of π ◦ fM (p) ([38, Lemma 3.7]), it follows that π ◦ fM and hence f provides a fibre bundle structure on N over Y . A point p of an Alexandrov space X is called an essential singular point if rad(Σp) ≤ π/2, where rad(Σp) = min ξ∈Σp max η∈Σp ∠(ξ, η) is the radius of Σp. Notice that if a point p ∈ X is not an essential singular point, then Σp is homeomorphic to a sphere ([18]) and a small metric ball around p is homeomorphic to Rn ([26, 27]), where n is the dimension of X. We also say that p is a topological singular point if Σp is not homeomorphic to a sphere. When no collapsing occurs, we have the following stability result. Theorem 2.4 (Stability Theorem [26]). Let a sequence of compact n-dimensional Alexandrov spaces Xi with curvature ≥ −1 converge to a compact Alexandrov space X of dimension n. Then Xi is homeomorphic to X for sufficiently large i. Part 1. Analyzing collapsed three-manifolds 3. Key lemma Let a sequence of pointed complete n-dimensional Rieman... |

76 |
Nilpotent Structures and Invariant Metrics on Collapsed Manifolds
- Cheeger, Fukaya, et al.
- 1992
(Show Context)
Citation Context ...s easily verified that as " → 0, (N/Γ, g) collapses to a closed interval under lim→0 |Kg |= 0. Corollary 7.4. Let Mi be a convergent sequence in M(3, D) of closed orientable three-dimensional Riemannian manifolds with finite fundamental groups. Suppose that the limit X of Mi has boundary as an Alexandrov space. Then Mi is homeomorphic to either S3, a lens space or a prism manifold. Proof. This follows from the discussion above and Corollary 0.4. q.e.d. 8. Comparison with bounded curvature collapsing In the bounded curvature case, the collapsing phenomena are well understood (see [12], [11], [8], etc.). Since we know no reference for the following result however, we give a proof. Proposition 8.1. Let Mi, i = 1, 2, . . . , be a sequence of closed n-dimensional Riemannian manifolds with |K |≤ 1, diam(Mi) ≤ D converging to a space X of dimension (n − 1). Then: (1) Mi is a Seifert fibred space over X for large i. (2) If each Mi is orientable, then the Alexandrov space X has no boundary. Proposition 8.1 (2) explains a difference between the bounded curvature collapsing and the lower curvature collapsing. Proof. We follow an argument in [12]. Let B(r) denote the metric ball B(0, r;Rn). In ... |

68 |
Seifert manifolds
- Orlik
- 1972
(Show Context)
Citation Context ...smooth Riemannian manifold and converges to the closed interval [ 0, ] under K ≥ 0. Case (2-ii-b) The case other than Case (2-ii-a). In this case M is a Seifert fibred space over P 2, where the number r of singular fibres satisfies r ≤ 1. If r = 0, namely, ϕ(S1-factor) = S1-factor, ϕ(∂D2-factor) = ∂Mo-factor, then M P 2×S1 = P 3#P 3. Consider the union M = B ∪A ∪C, where B and C are as in Case (2-ii-a). Note that ϕ : T 2 → T 2 is an isometry in this case. Hence M is a smooth Riemannian manifold and converges to the closed interval [ 0, ] under K ≥ 0. If r = 1, it follows (see [24]) that M is a prism manifold. Case (3) (B, C) = (Mo ×S1,Mo ×S1). In this case, M is doubly covered by a T 2-bundle over S1. In particular M admits a geometric structure modelled on R3, Nil or Sol (see [29]). We show that every such M B ∪ϕ C actually admits a sequence of metrics collapsing to a closed interval with bounded curvature |K |≤ Λ. First consider the monodoromy matrix J ∈ SL(2, Z) induced by the homomorphism ϕ∗ on the first homology group of the torus. If J can be represented by ( 1 n 0 1 ) , 32 takashi shioya & takao yamaguchi then M is finitely covered by T 3 (n = 0) or a nilm... |

65 |
The Riemannian structure of Alexandrov spaces
- Otsu, Shioya
- 1994
(Show Context)
Citation Context ... Key words and phrases. the Gromov-Hausdorff convergence, Alexandrov spaces, topology of three-manifolds, Seifert fibred spaces. 1 2 takashi shioya & takao yamaguchi Some answers are known in the extremal cases: If dimX = 0, the fundamental group of Mi is almost nilpotent (Fukaya and Yamaguchi [13]) and if dimX = n, Mi is homeomorphic to X (Perelman [26, 27], cf. Grove, Petersen and Wu [19]). In particular, for the above problem it suffices to consider only the case of dimX ≤ n − 1, the so called collapsing case. If X has no singular points, then X is a C0-Riemannian manifold (Otsu and Shioya [25]), and the Fibration Theorem (Yamaguchi [37]) implies that Mi is a fibre bundle over X with almost nonnegatively curved fibre. Actually the Fibration Theorem still holds if X has only ‘weak’ singularities ([38]) in some sense. According to Perelman ([28]), it is also known that if X has no ‘bad’ singularities (precisely called extremal subsets), there is an isomorphism πk(Mi, Fi) πk(X) for homotopy groups, where Fi is a ‘general fibre’ and i is large enough compared with k. When dimX ≤ n−1 and X may have ‘bad’ singularities, no solution to Problem 0.1 is known as of now. In this paper we com... |

63 |
3-manifolds
- Hempel
- 1976
(Show Context)
Citation Context ...v2−1(D)/π (see Corollary 14.3). collapsing three-manifolds 21 5. Topology near boundary In this section, we consider the case when the limit space X is a two-dimensional Alexandrov space with boundary. The argument in the previous section shows that the part M ′i of Mi converging to a part X0 of X away from the boundary ∂X is a Seifert fibred space over X0. Hence the essential point of the proof is to describe the topology of Mi − M ′i . Actually we prove that it is homeomorphic to ∂X × D2. For reader’s convenience, we give the definition of twisted bundles over surfaces. For the details, see [21]. Let S1 = {z ∈ C ||z |= 1} and I = [0, 1]. A twisted S1-bundle Mo ×S1 over a Mobius band Mo is defined as the quotient space (S1× I ×S1)/τ , where τ is the involution of S1 × I × S1 defined by τ(eiθ, t, eiη) = (ei(θ+π), 1− t, e−iη). Let N be a non-orientable surface and N the orientable double cover of N with the nontrivial deck transformation σ on N . Then a twisted I-bundle N×I over N is defined as the quotient space (N × I)/τ , where τ is the involution of N × I defined by τ(x, t) = (σ(x), 1 − t). Note that ∂ Mo ×S1 T 2, ∂N×I N and Mo ×S1 K2×I, where K2 denotes a Klei... |

57 |
The fundamental groups of almost non-negatively curved manifolds
- Fukaya, Yamaguchi
- 1992
(Show Context)
Citation Context ...ture ≥ −1. We now assume that Mi itself converges to X, i is sufficiently large, and consider the following: Problem 0.1. Describe the topological structure of Mi by using the geometry and topology of X. Received October 22, 1999. 1991 Subject Classification. Primary 53C20, 53C23; Secondary 57N10, 57M99. Key words and phrases. the Gromov-Hausdorff convergence, Alexandrov spaces, topology of three-manifolds, Seifert fibred spaces. 1 2 takashi shioya & takao yamaguchi Some answers are known in the extremal cases: If dimX = 0, the fundamental group of Mi is almost nilpotent (Fukaya and Yamaguchi [13]) and if dimX = n, Mi is homeomorphic to X (Perelman [26, 27], cf. Grove, Petersen and Wu [19]). In particular, for the above problem it suffices to consider only the case of dimX ≤ n − 1, the so called collapsing case. If X has no singular points, then X is a C0-Riemannian manifold (Otsu and Shioya [25]), and the Fibration Theorem (Yamaguchi [37]) implies that Mi is a fibre bundle over X with almost nonnegatively curved fibre. Actually the Fibration Theorem still holds if X has only ‘weak’ singularities ([38]) in some sense. According to Perelman ([28]), it is also known that if X has no ‘bad... |

51 | A proof of the generalized Schoenflies theorem
- Brown
- 1960
(Show Context)
Citation Context ...n Lemma 12.1. Then for " < δ with δ ≤ δp and "/δ ≤ "p, consecutive use of Theorem 10.1 together with Lemma 12.1 provides neighborhoods U ⊃ V of p and a gradient flow ψ on U − V for f satisfying the conditions (1), (2) and (4) in Definition 12.2. For the proof, it suffices to extend ψ to a pseudo-gradient flow on U for f. By Proposition 11.3, we may assume that U ⊃ V satisfy (e). Let E0 and E1 be the two component of ∂V − h((intB(p, δ) ∩ ∂C) × ∂(I ′ × J ′)) ⊂ ∂B(p, δ). Since ∂V S2, it follows from Proposition 11.3 and the generalized Schoenflies Theorem 54 takashi shioya & takao yamaguchi ([3]) that V I ′×J ′× [ 0, 1 ], where Ei corresponds to I ′×J ′×{i}. Let φi be a flow on Ei starting from {s′1}×J ′×{i}, reaching {s′0}×J ′×{i} and extending ψ, i = 0, 1. Then by Lemma 12.4, we have a flow φ on V extending φi and ψ restricted to ∂V ∩ intB(p, δ). The flow defined by the union of ψ and φ gives a required flow on U . q.e.d. Proposition 12.5. There exist an "0 > 0, a neighborhood U of ∂C and a pseudo-gradient flow ψ on U for f0 together with a homeomorphism h : ∂C × I × J → U such that for each x ∈ ∂C: (1) h({x} × I × ∂J) gives flow curves of ψ, that is, ψ(h(x, s1, tj), s) = h(x, s... |

39 | Deformation of homeomorphisms on stratified sets
- Siebenmann
- 1972
(Show Context)
Citation Context ... → N . The map ϕ is a product chart about U for π, and the image ϕ(U × N) is a product neighborhood around p. A finite dimensional topological space Y is said to be a WCS-space if it satisfies the following (1) and (2): (1) Y is a stratified space, i.e., it has a stratification Y ⊃ · · · ⊃ Y (n) ⊃ · · · ⊃ Y −1 = φ, such that Y (n)−Y (n−1) is a topological n-manifold without boundary. collapsing three-manifolds 45 (2) For each x ∈ Y (n) − Y (n−1) there is a cone C with vertex v and a homeomorphism ρ : Rn × C → Y onto an open neighborhood of x in Y such that ρ−1(Y (n)) = Rn × {v}. Theorem 10.1 ([33]). Let π : E → B be a topological submersion, and F = π−1(x0) the fibre over a point x0. We assume that F is a WCS-space. (1) For given compact sets A1, A2 of F and for open neighborhoods Ui of Ai in F , let ϕi : Ui × Ni → E be product charts about Ui for π. Then there exists a product chart ϕ : U ×N → E about an open set U ⊃ A1 ∪ A2 in F such that ϕ = { ϕ1 near A1 × {x0}, ϕ2 near (A2 − U1)× {x0}. (2) If π is proper in addition, then F ↪→ E π→ B is a locally trivial fibre bundle. Let f = (f1, . . . , fm) : U → Rm be a map on an open set U of X defined by fi(x) = d(Ai, x) for compact subsets Ai... |

38 |
Collapsing and pinching under a lower curvature bound,
- Yamaguchi
- 1991
(Show Context)
Citation Context ... convergence, Alexandrov spaces, topology of three-manifolds, Seifert fibred spaces. 1 2 takashi shioya & takao yamaguchi Some answers are known in the extremal cases: If dimX = 0, the fundamental group of Mi is almost nilpotent (Fukaya and Yamaguchi [13]) and if dimX = n, Mi is homeomorphic to X (Perelman [26, 27], cf. Grove, Petersen and Wu [19]). In particular, for the above problem it suffices to consider only the case of dimX ≤ n − 1, the so called collapsing case. If X has no singular points, then X is a C0-Riemannian manifold (Otsu and Shioya [25]), and the Fibration Theorem (Yamaguchi [37]) implies that Mi is a fibre bundle over X with almost nonnegatively curved fibre. Actually the Fibration Theorem still holds if X has only ‘weak’ singularities ([38]) in some sense. According to Perelman ([28]), it is also known that if X has no ‘bad’ singularities (precisely called extremal subsets), there is an isomorphism πk(Mi, Fi) πk(X) for homotopy groups, where Fi is a ‘general fibre’ and i is large enough compared with k. When dimX ≤ n−1 and X may have ‘bad’ singularities, no solution to Problem 0.1 is known as of now. In this paper we completely solve it in the case when n = 3 and ... |

36 | A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters
- Fukaya
- 1988
(Show Context)
Citation Context ...ex two. It is easily verified that as " → 0, (N/Γ, g) collapses to a closed interval under lim→0 |Kg |= 0. Corollary 7.4. Let Mi be a convergent sequence in M(3, D) of closed orientable three-dimensional Riemannian manifolds with finite fundamental groups. Suppose that the limit X of Mi has boundary as an Alexandrov space. Then Mi is homeomorphic to either S3, a lens space or a prism manifold. Proof. This follows from the discussion above and Corollary 0.4. q.e.d. 8. Comparison with bounded curvature collapsing In the bounded curvature case, the collapsing phenomena are well understood (see [12], [11], [8], etc.). Since we know no reference for the following result however, we give a proof. Proposition 8.1. Let Mi, i = 1, 2, . . . , be a sequence of closed n-dimensional Riemannian manifolds with |K |≤ 1, diam(Mi) ≤ D converging to a space X of dimension (n − 1). Then: (1) Mi is a Seifert fibred space over X for large i. (2) If each Mi is orientable, then the Alexandrov space X has no boundary. Proposition 8.1 (2) explains a difference between the bounded curvature collapsing and the lower curvature collapsing. Proof. We follow an argument in [12]. Let B(r) denote the metric ball B(0,... |

30 |
Geometric finiteness theorems via controlled topology
- Grove, Petersen, et al.
- 1991
(Show Context)
Citation Context ...the following: Problem 0.1. Describe the topological structure of Mi by using the geometry and topology of X. Received October 22, 1999. 1991 Subject Classification. Primary 53C20, 53C23; Secondary 57N10, 57M99. Key words and phrases. the Gromov-Hausdorff convergence, Alexandrov spaces, topology of three-manifolds, Seifert fibred spaces. 1 2 takashi shioya & takao yamaguchi Some answers are known in the extremal cases: If dimX = 0, the fundamental group of Mi is almost nilpotent (Fukaya and Yamaguchi [13]) and if dimX = n, Mi is homeomorphic to X (Perelman [26, 27], cf. Grove, Petersen and Wu [19]). In particular, for the above problem it suffices to consider only the case of dimX ≤ n − 1, the so called collapsing case. If X has no singular points, then X is a C0-Riemannian manifold (Otsu and Shioya [25]), and the Fibration Theorem (Yamaguchi [37]) implies that Mi is a fibre bundle over X with almost nonnegatively curved fibre. Actually the Fibration Theorem still holds if X has only ‘weak’ singularities ([38]) in some sense. According to Perelman ([28]), it is also known that if X has no ‘bad’ singularities (precisely called extremal subsets), there is an isomorphism πk(Mi, Fi) πk(X... |

25 |
A generalization of Berger’s rigidity theorem for positively curved manifolds,
- Gromoll, Grove
- 1987
(Show Context)
Citation Context ...sphere. From Proposition 9.10 and the proof of Lemma 10.3, we know that diam(Σp) > π/2 for every point p ∈ X − ∂C and hence Σp is homeomorphic to a sphere ([26]). Note that from the basic construction, diam(Σp) ≥ π/2 for every p ∈ C (if dimC ≥ 1). Thus for the proof of Proposition 11.3, we only have to care a point p ∈ ∂C with diam(Σp) = π/2. Let Σ be a two-dimensional compact Alexandrov surface with curvature ≥ 1 and without boundary. Suppose that diam(Σ) = π/2. For a subset B ⊂ Σ such that B = {x ∈ Σ |d(B, x) = π/2} is non-empty, we consider A1 = B and A2 = A1. Then we have A2 = A1 (see [15] for details). Proposition 11.4. Let Σ, A1 and A2 be as above. Then we have: (1) If both A1 and A2 are contractible, then Σ is homeomorphic to a sphere. (2) If one of A1 and A2 is not contractibe, then Σ is isometric to the projective plane, (the spherical suspension over S1 )/Z2, where the length of the circle S1 is less than or equal to 2π and Σ has constant curvature K = 1 outside the possible singular vertex. Proof. First we note that from the Alexandrov convexity: (a) Ai are convex sets. (b) Any distinct three points x, y ∈ Ai and z ∈ Aj , i = j span a geodesic triangle isometric to ... |

23 |
An introduction to the geometry of Alexandrov spaces
- Shiohama
- 1993
(Show Context)
Citation Context ...a′ = d(p′, s′), α′ = ∠p′s′q′, θ′ = ∠q′p′s′, and θ = ∠qps = π/2− α. Since ∠pqs = π/2, we have a′ ≥ a = d(p, s). It follows from the concavity of f that f ′(0) ≥ t a ≥ t a′ . Thus from the previous assertion, we obtain that α ≥ α′ and θ ≤ θ′.(9.1) Consider now a comparison triangle pqs in R2 and put θ = ∠qps, α = ∠psq. Since we may assume for our purpose that t > b, it follows from an obvious consideration with a′ ≥ a > t that α′ ≤ α ≤ π/2, θ′ ≤ θ and hence θ′ = θ = θ, α′ = α = α, a = a′ and ∠pqs = π/2.(9.2) 44 takashi shioya & takao yamaguchi It follows from the rigidity argument(cf.[30]) that pqs spans a totally geodesic flat triangle isometric to pqs. Furthermore, f ′(0) = t/a. It follows from the concavity of f that f(u) = tu/a for all u. Let xu and yu be the points on ∂Xt and qs respectively such that f(u) = d(γ(u), xu) and d(q, yu) = ub/a. Then it follows together with the comparison argument that d(xu, yu) ≤ d(xu, γ(u)) + d(γ(u), yu) ≤ t. Thus γ lies on the minimal connections from the points of qs to ∂Xt. By repeating the argument above for xu, yu, r, s in place of p, q, r, s, we conclude that the set of minimal connections xuyu, 0 ≤ u ≤ a, provides a totally geodes... |

15 |
A compactification of a manifold with asymptotically nonnegative curvature,
- Kasue
- 1988
(Show Context)
Citation Context ... nonnegative curvature, ( 1RY, y0) converges to the limit cone of Y as R → ∞. Therefore, for a µ > 1 and a sufficiently large R > 0, the rescaled annulus 1 Rδi A(pi;µ−1Rδi, µRδi) in Mi is dGH -close to the annulus A(o∞;µ−1, µ) in the limit cone for i large, where o∞ is the vertex of the limit cone. This together with a standard argument of critical point theory proves B(pi, Rδi) B(Si, Rδi). This completes the proof. q.e.d. Next we shortly discuss the ideal boundary of Y , where Y is as in Key Lemma 3.6. The ideal boundary Y (∞) of Y and the Tits metric ∠∞ on Y (∞) were defined in [2] (cf. [22], [32]). Let K be the asymptotic cone of Y defined as the pointed Gromov-Hausdorff limit of ("Y, y0) as " → 0 for a point y0 ∈ Y . Then K is the Euclidean cone over (Y (∞), ∠∞). collapsing three-manifolds 17 Lemma 3.7. There is an expanding map Σp → Y (∞). Proof. For a fixed " > 0 and for each x ∈ ∂B(p, "), take a minimal geodesic γi from pi to xi, where xi is a point in ∂B(pi, ") converging to x. Passing to a subsequence, we may assume that 1δi γi converges to a geodesic ray γx from y0 under the convergence ( 1δi B(pi, r), pi) → (Y, y0). Thus we have a map ϕ : ∂B(p, ") → Y (∞) defined by ... |

7 |
The Gaussian curvature of Alexandrov surfaces,
- Machigashira
- 1998
(Show Context)
Citation Context ...e define the rotation κ(σ) of σ with respect to the chosen side in the same manner. Let denote the open disk domain bounded by a triangle in X. The total curvature (or excess) ω() of in X is defined by ω() := α + β + γ − π, where α, β, γ are the inner angles of at its three vertices. Let D ⊂ intX be a relatively compact polygonal open region and find a triangulation of D with triangles {}. Then, the total curvature (or total excess) ω(D) of D is defined by ω(D) := ∑ ω() + ∑ x∈V ∩intD (2π − L(Σx)), where V is the set of vertices of the triangulation of D. According to [1] (see also [23]), the total curvature ω is independent of the triangulation and extends to a signed Radon measure over X with the following properties: (1) For any D ⊂ intX as above, we have the Gauss-Bonnet formula: ω(D) + κ(∂D) = 2πχ(D), where χ(D) denotes the Euler characteristic of D. (2) Any H2-measurable subset of X is ω-measurable and we have ω ≥ aH2, so that ω − aH2 is a (nonnegative) Radon measure. collapsing three-manifolds 61 (3) The restriction of ω onto ∂X is ω|∂X = 0. (4) For any minimal segment xy in X, ω|xy−{x,y} = 0. (5) For any x ∈ intX, ω({x}) = 2π − L(Σx). We now introduce the rotation me... |

7 |
of rays in Alexandrov spaces of nonnegative curvature,
- Mass
- 1994
(Show Context)
Citation Context ...gative curvature, ( 1RY, y0) converges to the limit cone of Y as R → ∞. Therefore, for a µ > 1 and a sufficiently large R > 0, the rescaled annulus 1 Rδi A(pi;µ−1Rδi, µRδi) in Mi is dGH -close to the annulus A(o∞;µ−1, µ) in the limit cone for i large, where o∞ is the vertex of the limit cone. This together with a standard argument of critical point theory proves B(pi, Rδi) B(Si, Rδi). This completes the proof. q.e.d. Next we shortly discuss the ideal boundary of Y , where Y is as in Key Lemma 3.6. The ideal boundary Y (∞) of Y and the Tits metric ∠∞ on Y (∞) were defined in [2] (cf. [22], [32]). Let K be the asymptotic cone of Y defined as the pointed Gromov-Hausdorff limit of ("Y, y0) as " → 0 for a point y0 ∈ Y . Then K is the Euclidean cone over (Y (∞), ∠∞). collapsing three-manifolds 17 Lemma 3.7. There is an expanding map Σp → Y (∞). Proof. For a fixed " > 0 and for each x ∈ ∂B(p, "), take a minimal geodesic γi from pi to xi, where xi is a point in ∂B(pi, ") converging to x. Passing to a subsequence, we may assume that 1δi γi converges to a geodesic ray γx from y0 under the convergence ( 1δi B(pi, r), pi) → (Y, y0). Thus we have a map ϕ : ∂B(p, ") → Y (∞) defined by ϕ(x) ... |

3 | Behavior of distant maximal geodesics in finitely connected complete 2-dimensional Riemannian manifolds, - Shioya - 1994 |

1 |
Alexandrov, Die innere Geometrie der konvexen Flachen, Akademie,
- D
- 1955
(Show Context)
Citation Context ...i=1 is called an (n, δ)-strainer at p ∈ X if it satisfies ∠aipbi > π − δ, ∠aipaj > π/2− δ, ∠bipbj > π/2− δ, ∠aipbj > π/2− δ, for every i = j. The number min {d(ai, p), d(bi, p) |1 ≤ i ≤ n} is called the length of the strainer. Let Xδ denote the set of (n, δ)-strained points of X. This has the structure of a Lipschitz n-manifold. Note that every point in Xδ has a small neighborhood almost isometric to an open subset of Rn for small δ. The boundary ∂X of X is inductively defined as the set of points p such that Σp has non-empty boundary. In dimension two, we have the following result ([4], [1]). Theorem 2.1. Any two-dimensional Alexandrov space X with curvature bounded below is a topological manifold possibly with boundary. Furthermore for every positive number δ, the set of interior points (resp. boundary points) of X at which the length of the space of directions is smaller than 2π − δ (resp. π − δ) is discrete. Next we recall the Fibration Theorem from [37], [38]. Let X be an Alexandrov space. The δ-strain radius at a point p ∈ Xδ is defined as the supremum of those r > 0 that there exists an (n, δ)-strainer at p of length r. The δ-strain radius of a closed domain Y ⊂ Xδ is, by ... |

1 | surfaces, Interscience Tracts - Convex - 1958 |

1 |
Hautes Etudes Sci.
- Volume, cohomology, et al.
- 1982
(Show Context)
Citation Context ...cf. [35]) that there are eight geometric structures of three-manifolds modelled on S3, R3, H3, S2×R1, H2×R, SL2(R), Nil and Sol. Corollary 0.9. For any D > 0, there exists a constant " = "(D) > 0 such that if a closed, prime three-manifold with infinite fundamental group admits a Riemannian metric contained in M(3, D) with volume < ", then it admits a geometric structure modelled on one of the seven geometries except H3. collapsing three-manifolds 5 Observe that a hyperbolic manifold M does not collapse under K ≥ −1 because of the non-vanishing property ‖M‖ = 0 for simplicial volume (Gromov [17], Thurston [34]). Combining our results above, we obtain the following corollary on the existence of geometric structures for the elements of M(3, D). Corollary 0.10. All elements but finitely many homeomorphism classes in M(3, D) admit geometric structures. In the proofs of our results, we essentially use a critical pointrescaling argument to understand the topology of a small neighborhood of Mi converging to a small neighborhood of a singular point of X. When dimX = 2, as the limit space of the rescaled Mi, we have a three-dimensional complete open nonnegatively curved Alexandrov space Y whi... |

1 |
Edited by Silvio Levy.
- geometry, topology
- 1997
(Show Context)
Citation Context ...he case dimX = 0 previously stated in the following way. Corollary 0.7. Suppose that X is a point. Then a finite cover of Mi is homeomorphic to S1×S2, T 3, a nilmanifold or a simply connected Alexandrov space with nonnegative curvature. Conjecture 0.8. Any three-dimensional compact, simply connected, nonnegatively curved Alexandrov space without boundary which is a topological manifold is homeomorphic to a sphere. If Conjecture 0.8 is solved, everything will be clear about Problem 0.1 for n = 3. The above conjecture is certainly true in the Riemannian case ([20]). It is known by Thurston (cf. [35]) that there are eight geometric structures of three-manifolds modelled on S3, R3, H3, S2×R1, H2×R, SL2(R), Nil and Sol. Corollary 0.9. For any D > 0, there exists a constant " = "(D) > 0 such that if a closed, prime three-manifold with infinite fundamental group admits a Riemannian metric contained in M(3, D) with volume < ", then it admits a geometric structure modelled on one of the seven geometries except H3. collapsing three-manifolds 5 Observe that a hyperbolic manifold M does not collapse under K ≥ −1 because of the non-vanishing property ‖M‖ = 0 for simplicial volume (Gromov [17], Th... |