Generalized polar transforms of spacelike isothermic surfaces
Abstract
In this paper, we generalize the polar transforms of spacelike isothermic surfaces in Q41 to n-dimensional pseudo-Riemannian space forms Qnr. We show that there exist c-polar spacelike isothermic surfaces derived from a spacelike isothermic surface in Qnr, which are into Sn+1r(c), Hn+1r-1(c) or Qnr 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.