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

We provide the basic theory on the conformal geometry of timelike surfaces in pseudo-Riemannian space forms, which is to generalize the fundamental work of Burstall et al. for spacelike surfaces. Then we have 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) x O(1. 1)-type curved flat. Finally two permutability theorems of c-polar transforms are established.