• Previous Article
    Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $
  • ERA Home
  • This Issue
  • Next Article
    Finite/fixed-time synchronization for complex networks via quantized adaptive control
doi: 10.3934/era.2021005

Tori can't collapse to an interval

018 McAllister Bldg, University Park, PA 16802-6402, USA

* Corresponding author: Sergio Zamora

Received  October 2020 Published  January 2021

Fund Project: The author would like to thank Raquel Perales and Anton Petrunin

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. Electronic Research Archive, doi: 10.3934/era.2021005
References:
[1]

L. Auslander and M. Kuranishi, On the holonomy group of locally Euclidean spaces, Ann. of Math., 65 (1957), 411-415.  doi: 10.2307/1970053.  Google Scholar

[2] R. L. Bishop and R. J. Crittenden, Geometry of manifolds, Academic Press, 1964.   Google Scholar
[3]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.  Google Scholar

[4]

Y. BuragoM. Gromov and G. A. D. Perel'man, Alexandrov spaces with curvature bounded below, Russian Mathematical Surveys, 47 (1992), 1-58.  doi: 10.1070/RM1992v047n02ABEH000877.  Google Scholar

[5]

J. W. S. Cassels, An Introduction to The Geometry Of Numbers, Springer Science & Business Media (2012). Google Scholar

[6]

L. S. Charlap, Bieberbach Groups and Flat Manifolds, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8687-2.  Google Scholar

[7]

M. Gromov, Filling Riemannian manifolds, J. Differential Geom., 18 (1983), 1-147.  doi: 10.4310/jdg/1214509283.  Google Scholar

[8]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.  doi: 10.1007/BF02698687.  Google Scholar

[9]

M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser Boston, Inc., Boston, MA, 2007.  Google Scholar

[10]

M. Gromov and H. B. Lawson Jr, Spin and scalar curvature in the presence of a fundamental group Ⅰ, Ann. of Math., 111 (1980), 209-230.  doi: 10.2307/1971198.  Google Scholar

[11]

V. Kapovitch, Perelman's stability theorem, Surveys in Differential Geometry, 11 (2006), 103–136. arXiv: math/0703002. doi: 10.4310/SDG.2006.v11.n1.a5.  Google Scholar

[12]

V. Kapovitch, Restrictions on collapsing with a lower sectional curvature bound, Math. Z., 249 (2005), 519-539.  doi: 10.1007/s00209-004-0715-3.  Google Scholar

[13]

V. KapovitchA. Petrunin and W. Tuschmann, Almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Ann. of Math., 171 (2010), 343-373.  doi: 10.4007/annals.2010.171.343.  Google Scholar

[14]

M. G. Katz, Torus cannot collapse to a segment, J. Geom., 111 (2020), Paper No. 13, 8 pp. doi: 10.1007/s00022-020-0525-8.  Google Scholar

[15]

G. Ya. Perelman, Alexandrov spaces with curvature bounded from below Ⅱ, Leningrad Branch of Steklov Institute. St. Petesburg (1991) Google Scholar

[16]

T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133 (1991), 317-357.  doi: 10.2307/2944340.  Google Scholar

show all references

References:
[1]

L. Auslander and M. Kuranishi, On the holonomy group of locally Euclidean spaces, Ann. of Math., 65 (1957), 411-415.  doi: 10.2307/1970053.  Google Scholar

[2] R. L. Bishop and R. J. Crittenden, Geometry of manifolds, Academic Press, 1964.   Google Scholar
[3]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.  Google Scholar

[4]

Y. BuragoM. Gromov and G. A. D. Perel'man, Alexandrov spaces with curvature bounded below, Russian Mathematical Surveys, 47 (1992), 1-58.  doi: 10.1070/RM1992v047n02ABEH000877.  Google Scholar

[5]

J. W. S. Cassels, An Introduction to The Geometry Of Numbers, Springer Science & Business Media (2012). Google Scholar

[6]

L. S. Charlap, Bieberbach Groups and Flat Manifolds, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8687-2.  Google Scholar

[7]

M. Gromov, Filling Riemannian manifolds, J. Differential Geom., 18 (1983), 1-147.  doi: 10.4310/jdg/1214509283.  Google Scholar

[8]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.  doi: 10.1007/BF02698687.  Google Scholar

[9]

M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser Boston, Inc., Boston, MA, 2007.  Google Scholar

[10]

M. Gromov and H. B. Lawson Jr, Spin and scalar curvature in the presence of a fundamental group Ⅰ, Ann. of Math., 111 (1980), 209-230.  doi: 10.2307/1971198.  Google Scholar

[11]

V. Kapovitch, Perelman's stability theorem, Surveys in Differential Geometry, 11 (2006), 103–136. arXiv: math/0703002. doi: 10.4310/SDG.2006.v11.n1.a5.  Google Scholar

[12]

V. Kapovitch, Restrictions on collapsing with a lower sectional curvature bound, Math. Z., 249 (2005), 519-539.  doi: 10.1007/s00209-004-0715-3.  Google Scholar

[13]

V. KapovitchA. Petrunin and W. Tuschmann, Almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Ann. of Math., 171 (2010), 343-373.  doi: 10.4007/annals.2010.171.343.  Google Scholar

[14]

M. G. Katz, Torus cannot collapse to a segment, J. Geom., 111 (2020), Paper No. 13, 8 pp. doi: 10.1007/s00022-020-0525-8.  Google Scholar

[15]

G. Ya. Perelman, Alexandrov spaces with curvature bounded from below Ⅱ, Leningrad Branch of Steklov Institute. St. Petesburg (1991) Google Scholar

[16]

T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133 (1991), 317-357.  doi: 10.2307/2944340.  Google Scholar

Figure 1.  Flat Klein bottles can converge to an interval
Figure 2.  The Fibration Theorem gives us a decomposition $ X_n = S_1 \# S_2 $
Figure 3.  The configuration $ (q_n; \tilde{p}_n ,a_n, b_n) $ violates the Alexandrov condition
[1]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[2]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

[3]

Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125

[4]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090

[5]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280

[6]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318

[7]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389

[8]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[9]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[10]

Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1093-1102. doi: 10.3934/dcdss.2020385

[11]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390

[12]

Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333

 Impact Factor: 0.263

Article outline

Figures and Tables

[Back to Top]