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Transactions of the American Mathematical Society

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The role of the mean curvature in the immersion theory of surfaces


Author: H. W. Alexander
Journal: Trans. Amer. Math. Soc. 47 (1940), 230-253
MSC: Primary 53.0X
DOI: https://doi.org/10.1090/S0002-9947-1940-0001621-7
MathSciNet review: 0001621
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References [Enhancements On Off] (What's this?)

  • [1] W. C. Graustein, Applicability with preservation of both curvatures, Bull. Amer. Math. Soc. 30 (1924), no. 1-2, 19–23. MR 1560835, https://doi.org/10.1090/S0002-9904-1924-03839-7
  • [2] The tensor $ {\varepsilon ^{\alpha \beta }}$ is defined and discussed in A. Duschek, Lehrbuch der Differentialgeometrie, Leipzig, Teubner, 1930, p. 99, formula (19).
  • [3] The conditions (5.1) correspond to the conditions ``$ z = Q/P$ satisfies the equations (4)'' of Theorem 1 in Graustein's paper in the Duke Mathematical Journal.
  • [4] A. R. Forsyth, Differential Geometry, Cambridge, University Press, 1920, p. 84.
  • [5] Luther Pfahler Eisenhart, Riemannian geometry, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Eighth printing; Princeton Paperbacks. MR 1487892
  • [6] See Theorem 1 in Graustein's paper in the Duke Mathematical Journal.
  • [7] Equation (6.11) is another form of the condition $ {\Delta _2}\log ( - K) - 4K = 0$, which is due to Ricci. See L. Bianchi, Vorlesungen über Differentialgeometrie, translated by M. Lukat, Leipzig, Teubner, 1899, p. 382.
  • [8] Goursat-Hedrick, A Course in Mathematical Analysis, vol. 2, part 2, New York, Ginn, 1917, p. 209.
  • [9] H. Hamburger, Über Kurvenetze mit isolierten singularitäten auf geschlossenen Flächen, Mathematische Zeitschrift, vol. 19 (1924), pp. 50-56.
  • [10] Wilhelm Blaschke, Über die Geometrie von Laguerre, Math. Z. 24 (1926), no. 1, 617–621 (German). MR 1544781, https://doi.org/10.1007/BF01216800
  • [11] P. Franklin, Regions of positive and negative curvature on closed surfaces, Journal of Mathematics and Physics, vol. 13 (1934), pp. 253-260.
  • [12] G. Darboux, Théorie des Surfaces, Paris, Gauthier-Villars, 1896, vol. 4, pp. 448-465.
  • [13] Allvar Gullstrand, Zur Kenntniss der Kreispunkte, Acta Math. 29 (1905), no. 1, 59–100 (German). MR 1555011, https://doi.org/10.1007/BF02403199
  • [14] See, for example, S. Lefschetz, Topology, American Mathematical Society Colloquium Publications, vol. 12, New York, 1930, p. 44.

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DOI: https://doi.org/10.1090/S0002-9947-1940-0001621-7
Article copyright: © Copyright 1940 American Mathematical Society