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

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 (±)−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 (±)−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) × O(1, 1)−type curved flat. Finally two permutability theorems of c−polar transforms are established.