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

    Related Papers:

  • 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.



    加载中


    [1] On the holonomy group of locally Euclidean spaces. Ann. of Math. (1957) 65: 411-415.
    [2] (1964) Geometry of manifolds.Academic Press.
    [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] Alexandrov spaces with curvature bounded below. Russian Mathematical Surveys (1992) 47: 1-58.
    [5] J. W. S. Cassels, An Introduction to The Geometry Of Numbers, Springer Science & Business Media (2012).
    [6] L. S. Charlap, Bieberbach Groups and Flat Manifolds, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8687-2
    [7] Filling Riemannian manifolds. J. Differential Geom. (1983) 18: 1-147.
    [8] Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. (1981) 53: 53-73.
    [9] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser Boston, Inc., Boston, MA, 2007.
    [10] Spin and scalar curvature in the presence of a fundamental group Ⅰ. Ann. of Math. (1980) 111: 209-230.
    [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] Restrictions on collapsing with a lower sectional curvature bound. Math. Z. (2005) 249: 519-539.
    [13] Almost nonnegative curvature, and the gradient flow on Alexandrov spaces. Ann. of Math. (2010) 171: 343-373.
    [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)
    [16] Collapsing and pinching under a lower curvature bound. Ann. of Math. (1991) 133: 317-357.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1036) PDF downloads(152) Cited by(0)

Article outline

Figures and Tables

Figures(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog