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I have been reading Road to Reality by Roger Penrose, but cant go beyond chapter 8. I do not understand why topological equivalence does not imply conformal equivalence. In particular I cannot really make sense of his argument as to why a thin torus is not conformally the same as a fat one something he refers to as "pretty clear". Another source of confusion is that as far as I know any two parallelograms can be transformed into each other by a holomorphic transformation (Riemann mapping theorem). Doesnt this imply that the tori generated by these parallelograms should be conformally equivalent? Please help.

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# Conformal equivalence

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