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The Mathematical Intelligencer

, Volume 16, Issue 4, pp 31–37 | Cite as

A gallery of constant-negative-curvature surfaces

  • Robert McLachlan
Article

Keywords

Soliton Coordinate Line Asymptotic Direction Constant Negative Curvature Asymptotic Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Method for solving the sine-Gordon equation,Phys. Rev. Lett. 30 (1973), 191–193.CrossRefMathSciNetGoogle Scholar
  2. 2.
    M. J. Ablowitz and H. Segur,Solitons and the Inverse Scattering Transform, Philadelphia: SIAM, (1981).CrossRefzbMATHGoogle Scholar
  3. 3.
    Visualizations were done in geomview, available from the Geometry Center, Minneapolis, by anonymous ftp to geom.umn.edu.Google Scholar
  4. 4.
    L. P. Eisenhart,A Treatise on the Differential Geometry of Curves and Surfaces, New York: Dover (1960) (originally published 1909).zbMATHGoogle Scholar
  5. 5.
    Gerd Fischer,Mathematische Modelle, Braunschweig Wiesbaden Vieweg, 2 vols. (1986).Google Scholar
  6. 6.
    H. Hasimoto, A soliton on a vortex filament,J. Fluid Mech. 51 (1972), 477–485.CrossRefzbMATHGoogle Scholar
  7. 7.
    J. N. Hazzidakis, Ueber einige Eigenschaften der Flächen mit constantem Krümmungsmaass,J. reine angew. Math. (“Crelle’s Journal”) 88 (1880), 68–73.Google Scholar
  8. 8.
    D. Hilbert, Ueber Flächen von constanter Gaussscher Krümmung,Trans. Amer. Math. Soc. 2 (1901), 87–99.zbMATHMathSciNetGoogle Scholar
  9. 9.
    S. B. Kadomtsev, Surfaces with constant exterior geometry of negative curvature,Math. Notes Acad. Sci. USSR 47(4) (1990), 339–341.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    G. L. Lamb Jr.,, Solitons on moving space curves,J. Math. Phys. 18(8) (1977), 1654–1661.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    M. Melko and I. Sterling, Application of soliton theory to the construction of pseudospherical surfaces in R3,Ann. Global Anal. Geom. 11(1) (1993), 65–107.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    K. Nakayama, H. Segur, and M. Wadati, Integrability and the motion of curves,Phys. Rev. Lett. 69 (1992), 2603–2606.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    A. Seeger, H. Donth, and A. Kochendörfer, Théorie der Versetzungen in eindimensionalen Atomreihen. III: Ver- setzungen, Eigenbewegungen und ihre Wechselwirkung, Z.Phys. 134 (1953), 173–193.CrossRefzbMATHGoogle Scholar
  14. 14.
    H. Segur, Who cares about integrability?,Physica D 51 (1991), 343–359.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    H. Segur, personal communication.Google Scholar
  16. 16.
    J. J. Stoker,Differential Geometry, New York: Wiley- Interscience (1969).zbMATHGoogle Scholar
  17. 17.
    M. Spivak,A Comprehensive Introduction to Differential Geometry, Volume III, Boston: Publish or Perish, Inc. (1975).Google Scholar
  18. 18.
    P. L. Tchebychev,Sur la coupe des větements(1878), (Œuvres, vol. II, New York: Chelsea (1962), 708.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 1994

Authors and Affiliations

  • Robert McLachlan
    • 1
  1. 1.University of Colorado at BoulderBoulderUSA

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