Generalized polar transforms of spacelike isothermic surfaces

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In this paper, we generalize the polar transforms of spacelike isothermic surfaces in Q14 to n-dimensional pseudo-Riemannian space forms Qrn. We show that there exist c-polar spacelike isothermic surfaces derived from a spacelike isothermic surface in Qrn, which are into Srn+1(c), Hr-1n+1(c) or Qrn depending on c > 0, < 0, or = 0. The c-polar isothermic surfaces can be characterized as generalized H-surfaces with null minimal sections. We also prove that if both the original surface and its c-polar surface are closed immersion, then they have the same Willmore functional. As examples, we discuss some product surfaces and compute the c-polar transforms of them. In the end, we derive the permutability theorems for c-polar transforms and Darboux transform and spectral transform of isothermic surfaces.

Original languageEnglish
Pages (from-to)403-411
Number of pages9
JournalJournal of Geometry and Physics
Issue number2
StatePublished - Feb 2012
Externally publishedYes


  • Christoffel transform
  • Darboux transform
  • Permutability theorem
  • Spacelike isothermic surfaces
  • Spectral transform
  • c-polar transform


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