doi: 10.3934/era.2021005

Tori can't collapse to an interval

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* 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

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

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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

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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
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