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Flat Klein bottles can converge to an interval
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doi: 10.3934/era.2021005
Tori can't collapse to an interval
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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.
Mathematics Subject Classification:
Primary: 53C23, 53C20; Secondary: 53C21.
Citation:
Sergio Zamora.
Tori can't collapse to an interval. Electronic Research Archive, doi: 10.3934/era.2021005
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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. |
| [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. |
| [4] |
Y. Burago, M. Gromov and G. A. D. Perel'man,
Alexandrov spaces with curvature bounded below, Russian Mathematical Surveys, 47 (1992), 1-58.
doi: 10.1070/RM1992v047n02ABEH000877. |
| [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. |
| [7] |
M. Gromov,
Filling Riemannian manifolds, J. Differential Geom., 18 (1983), 1-147.
doi: 10.4310/jdg/1214509283. |
| [8] |
M. Gromov,
Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.
doi: 10.1007/BF02698687. |
| [9] |
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser Boston, Inc., Boston, MA, 2007. |
| [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. |
| [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. |
| [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. |
| [13] |
V. Kapovitch, A. 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. |
| [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. |
| [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. |
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. |
| [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. |
| [4] |
Y. Burago, M. Gromov and G. A. D. Perel'man,
Alexandrov spaces with curvature bounded below, Russian Mathematical Surveys, 47 (1992), 1-58.
doi: 10.1070/RM1992v047n02ABEH000877. |
| [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. |
| [7] |
M. Gromov,
Filling Riemannian manifolds, J. Differential Geom., 18 (1983), 1-147.
doi: 10.4310/jdg/1214509283. |
| [8] |
M. Gromov,
Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.
doi: 10.1007/BF02698687. |
| [9] |
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser Boston, Inc., Boston, MA, 2007. |
| [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. |
| [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. |
| [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. |
| [13] |
V. Kapovitch, A. 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. |
| [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. |
| [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. |
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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