I’m currently reading differential geometry in the large by H. Hopf. On page 99 Hopf writes “a simple proof of the possibility of introducing isothermal parameters can be given providing E, F, G are analytic, where the proof is given by continuing E, F, G into the complex domain”. Does anyone know what this simple proof is about? I tried looking on the internet and also tried myself, but I failed.
(Or at least how one could possibly continue E F G into the complex domain)
This article Isothermal coordinates has a link to a translation of Gauss's original paper of 1822. Gauss's proof is based on the idea of treating the $E,F,G$ as functions that can be represented as power series, and replacing the real arguments in those power series with complex values.Gauss may not have explicitly stated that however, since he is very terse.