Abstract
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 \({(\pm)}\)-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.
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Yuping Song: Supported by the Project 11401496 of NSFC, Natural Science Foundation of Fujian Province of China (Grant No. 2015J05015) and the Fundamental Research Funds for the Central Universities (Grant No. 20720140527).
Peng Wang: supported by the Project 11571255 of NSFC and the Fundamental Research Funds for the Central Universities.
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Song, Y., Wang, P. On transforms of timelike isothermic surfaces in pseudo-Riemannian space forms. Results Math 71, 1421–1442 (2017). https://doi.org/10.1007/s00025-016-0607-y
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DOI: https://doi.org/10.1007/s00025-016-0607-y