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

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- by H. W. Alexander PDF
- Trans. Amer. Math. Soc.
**47**(1940), 230-253 Request permission

## References

- W. C. Graustein,
*Applicability with preservation of both curvatures*, Bull. Amer. Math. Soc.**30**(1924), no. 1-2, 19–23. MR**1560835**, DOI 10.1090/S0002-9904-1924-03839-7
The tensor ${\varepsilon ^{\alpha \beta }}$ is defined and discussed in A. Duschek, - 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, - Wilhelm Blaschke,
*Über die Geometrie von Laguerre*, Math. Z.**24**(1926), no. 1, 617–621 (German). MR**1544781**, DOI 10.1007/BF01216800
P. Franklin, - Allvar Gullstrand,
*Zur Kenntniss der Kreispunkte*, Acta Math.**29**(1905), no. 1, 59–100 (German). MR**1555011**, DOI 10.1007/BF02403199
See, for example, S. Lefschetz,

*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.

*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.

*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.

*Topology*, American Mathematical Society Colloquium Publications, vol. 12, New York, 1930, p. 44.

## Additional Information

- © Copyright 1940 American Mathematical Society
- 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