Theoretical and Mathematical Physics

, Volume 128, Issue 1, pp 946–956 | Cite as

Integration of the Gauss–Codazzi Equations

  • V. E. Zakharov


The Gauss–Codazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in \(\mathbb{R}^{3} \) are integrable by the dressing method. This method allows constructing classes of Combescure-equivalent surfaces with the same “rotation coefficients.” Each equivalence class is defined by a function of two variables (“master function of a surface”). Each class of Combescure-equivalent surfaces includes the sphere. Different classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.


Equivalence Class Quadratic Form Function Zero Simple Class Master Function 
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Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • V. E. Zakharov
    • 1
    • 2
  1. 1.Landau Institute for Theoretical Physics, RASChernogolovkaRussia
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA

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