Polar transform of Spacelike isothermic surfaces in 4-dimensional Lorentzian space forms

TL;DR

This paper investigates the polar transform of spacelike conformal isothermic surfaces in 4D Lorentzian space forms, showing it preserves the isothermic property and relates to known transforms through permutability theorems.

Contribution

It introduces and analyzes the polar transform for spacelike isothermic surfaces, establishing its preservation of properties and connections with existing transforms.

Findings

Polar transform preserves spacelike conformal isothermic surfaces.

The transform is related to Darboux and spectral transforms.

Permutability theorems link the new and known transforms.

Abstract

The conformal geometry of spacelike surfaces in 4-dimensional Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the known transforms (Darboux transform and spectral transform) of isothermic surfaces by establishing the permutability theorems.