2013/07/09 by Song, Yuping, Wang, Peng
#53A30 #53B30 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1307.2515
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.