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I was reading this paper

https://arxiv.org/pdf/math/0106036.pdf

and it talks about conformal homeomorphism, can someone give me the definition please?

Claudio Delfino
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  • How much Riemannian geometry do you know? – Blazej Jan 31 '19 at 23:25
  • I know what is an homeomorphism and what is a conformal map, but i never faced the conformal homeomorphism – Claudio Delfino Feb 01 '19 at 01:39
  • What do would you say a conformal map is? –  Feb 01 '19 at 16:32
  • For me a conformal map is an $f\colon D \to D'$ , where $D,D' \subset \mathbb{C}$, $f$ analytic and one to one and onto (which should implies that $f'(z)\ne0$ for all z and $f^{-1}$ is a conformal map too) – Claudio Delfino Feb 01 '19 at 17:31
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    What you refer as a conformal map is a special case. If you have two Riemannian manifolds $M, N$ with metrics $g$ and $h$, conformal homeomorphism $M \to N$ would be a diffeomorphism $\phi : \ M \to N$ such that pullback of $h$ coincides with $g$ multiplied by some function. If $M$ and $N$ are open subsets of complex plane this is equivalent to the condition you quoted (provided that you restrict attention to orientation preserving maps; otherwise you have to include anti-holomorphic functions also). – Blazej Feb 01 '19 at 21:37
  • It depends on the author whether or not (in 2D) what you call a 'conformal map' is what they call a 'conformal homeomorphism'. Sometimes they drop the 1-1 requirement. –  Feb 02 '19 at 15:48

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