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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Isothermic surfaces and the Gauss map


Author: Bennett Palmer
Journal: Proc. Amer. Math. Soc. 104 (1988), 876-884
MSC: Primary 53C42
DOI: https://doi.org/10.1090/S0002-9939-1988-0964868-7
MathSciNet review: 964868
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Abstract: We give a necessary and sufficient condition for the Gauss map of an immersed surface $ M$ in $ n$-space to arise simultaneously as the Gauss map of an anti-conformal immersion of $ M$ into the same space. The condition requires that the lines of curvature of each normal section lie on the zero set of a harmonic function. The result is applied to a class of surfaces studied by S. S. Chern which admit an isometric deformation preserving the principal curvatures.


References [Enhancements On Off] (What's this?)

  • [1] S.-S. Chern, Deformation of surfaces preserving principal curvatures, Differential Geometry and Complex Analysis, H. E. Rauch Memorial Volume, Springer-Verlag, 1985, pp. 155-163. MR 780041 (86h:53005)
  • [2] L. P. Eisenhart, A treatise on the differential geometry of curves and surfaces, Dover, New York, 1909. MR 0115134 (22:5936)
  • [3] D. A. Hoffman and R. Osserman, The Gauss map of a surface in $ {{\mathbf{R}}^n}$, J. Differential Geom. 18 (1983), 733-754. MR 730925 (85i:53059)
  • [4] -, The Gauss map of surfaces in $ {{\mathbf{R}}^3}$ and $ {{\mathbf{R}}^4}$, Proc. London Math. Soc. (3) 50 (1985), 27-56. MR 765367 (86f:58034)
  • [5] H. Hopf, Lectures on differential geometry in the large, Lecture Notes in Math., vol. 1000, Springer-Verlag, Berlin and New York, 1984. MR 707850 (85b:53001)
  • [6] K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), 89-99. MR 552581 (81c:53005b)

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DOI: https://doi.org/10.1090/S0002-9939-1988-0964868-7
Article copyright: © Copyright 1988 American Mathematical Society

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