On transforms of timelike isothermic surfaces in pseudo-Riemannian space forms

TL;DR

This paper develops the conformal geometry framework for timelike surfaces in pseudo-Riemannian space forms and explores their transforms, including polar, Darboux, and spectral transforms, revealing preservation properties and permutability theorems.

Contribution

It introduces a conformal geometric theory for timelike surfaces and characterizes their transforms, extending classical results to the pseudo-Riemannian setting.

Findings

$c$-polar transforms preserve timelike isothermic surfaces

Darboux pairs are characterized as Lorentzian curved flats

Permutability theorems for $c$-polar transforms are established

Abstract

The basic theory on the conformal geometry of timelike surfaces in pseudo-Riemannian space forms is introduced, which is parallel to the classical framework of Burstall etc. for spacelike surfaces. Then we provide a discussion on the transforms of timelike $(\pm)-$isothermic surfaces (or real isothermic, complex isothermic surfaces), including $c-$ polar transforms, Darboux transforms and spectral transforms. The first main result is that $c-$polar transforms preserve timelike $(\pm)-$isothermic surfaces, which are generalizations of the classical Christoffel transforms. The next main result is that a Darboux pair of timelike isothermic surfaces can also be characterized as a Lorentzian $O(n-r+1,r+1)/O(n-r,r)\times O(1,1)-$type curved flat. Finally two permutability theorems of $c-$polar transforms are established.