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

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.