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Transactions of the American Mathematical Society

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Inverting a cylinder through isometric immersions and isometric embeddings


Authors: B. Halpern and C. Weaver
Journal: Trans. Amer. Math. Soc. 230 (1977), 41-70
MSC: Primary 58D10; Secondary 57D40
DOI: https://doi.org/10.1090/S0002-9947-1977-0474388-1
MathSciNet review: 0474388
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Abstract: It is shown that a right circular cylinder can be turned inside out through immersions which preserve its flat Riemannian metric if and only if its diameter is greater than its height. Such a cylinder can be turned inside out through embeddings which preserve its flat Riemannian metric provided its diameter is greater than $ (\pi + 2)/\pi $ times its height. A flat Möbius strip has an immersion into Euclidean three dimensional space which preserves its Riemannian metric if and only if its length is greater than $ \pi /2$ times its height.


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  • [1] S. Barr, Experiments in topology, Crowell, New York, 1964.
  • [2] S.-S. Chern and R. K. Lashof, On the total curvature of immersed manifolds, Amer. J. Math. 79 (1957), 306-318. MR 18, 927. MR 0084811 (18:927a)
  • [3] S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214-222. MR 8, 51. MR 0016676 (8:51a)
  • [4] M. L. Gromov and V. A. Rohlin, Imbeddings and immersions in Riemannian geometry, Uspehi Mat. Nauk 25 (1970), no. 5 (155), 3-62 = Russian Math. Surveys 25 (1970), no. 5, 1-57. MR 44 #7571. MR 0290390 (44:7571)
  • [5] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901-920. MR 23 #A4106. MR 0126812 (23:A4106)
  • [6] D. Laugwitz, Differential and Riemannian geometry, Academic Press, New York, 1965. MR 30 #2406. MR 0172184 (30:2406)
  • [7] M. Sadowsky, Ein elementarer Beweis für die Existenz eines abwickelbaren Möbiusschen Bands und Zurückführung des geometrischen Problems auf ein Variationsproblem, Sitzgber. Preuss. Akad. Wiss. 22 (1930), 412-415.
  • [8] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [9] J. J. Stoker, Jr., Differential geometry, Interscience, New York, 1969. MR 39 #2072. MR 0240727 (39:2072)
  • [10] H. Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276-284. MR 1556973
  • [11] W. Wunderlich, Über ein abwickelbares Möbiusband, Monatsh. Math. 66 (1962), 276-289. MR 26 #680. MR 0143115 (26:680)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0474388-1
Keywords: Isometric immersion, isometric embedding, homotopy, isotopy, Möbius band, flat cylinder
Article copyright: © Copyright 1977 American Mathematical Society

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