Generalized polar transforms of spacelike isothermic surfaces

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