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

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Article copyright: © Copyright 1940 American Mathematical Society