I'm reading a book and found a sentence I don't understand:

Every Riemann surface $S$ supports an essentially unique complete metric of constant curvature $1$, $0$ or $-1$. Every point of $S$ has a neighborhood $U$ which is isometrically isomorphic to an open set in the Riemann sphere ($\mathbb{P}$), the complex plane ($\mathbb{C}$) or the open unit disc ($\mathbb{D}$).

Speciffically my doubt is: What does "isometrically isomorphic" means in this context? I searched in the web and found something about vector spaces, but the book has not mentioned vector spaces so far so i'm confused.

  • $\begingroup$ This simply means isometric, that is, there exists a diffeomorphism preserving Riemannian metrics $\endgroup$ – Moishe Kohan Oct 17 '13 at 14:58
  • $\begingroup$ Where can I get more information about the result mentioned in the sentence? $\endgroup$ – antony almeida Oct 17 '13 at 15:08
  • $\begingroup$ "isometrically isomorphic" only makes sense when the metric is specified on both sides. On ${\mathbb P}$ the usual spherical metric is meant, on ${\mathbb C}$ the euclidean metric, and on ${\mathbb D}$ the metric $ds={|dz|\over 1-|z|^2}$. $\endgroup$ – Christian Blatter Oct 17 '13 at 15:10
  • $\begingroup$ almeida: For references, google "uniformization theorem". Your book is also missing that the constant curvature metric is conformal with respect to the given complex structure. $\endgroup$ – Moishe Kohan Oct 17 '13 at 15:41

The use of the term "isometrically isomorphic" is not correct in this context. "Isomorphic" is used in the context when there are algebraic structures such as sums, products, compositions, etc. It should be simply "isometric". Note that the two claims are separate mathematical theorems, the first one being a stronger global result, whereas the second one is purely local.

  • $\begingroup$ "Isomorphic" can be used in any context to mean preserving all relevant structure. "Isometrically isomorphic" is perhaps a hint that this author uses "isometric" to mean locally isometric. $\endgroup$ – Anthony Carapetis Oct 17 '13 at 15:23
  • $\begingroup$ Do you have a source for using "isomorphic" in this sense? $\endgroup$ – Mikhail Katz Oct 18 '13 at 7:14
  • $\begingroup$ The Wikipedia pages for homeomorphism and isometry list "topological isomorphism" and "isometric isomorphism" respectively as alternative names. $\endgroup$ – Anthony Carapetis Oct 18 '13 at 7:26
  • $\begingroup$ Wiki is not a reliable source. In fact it may be the source of the incorrect usage reported by the OP. $\endgroup$ – Mikhail Katz Oct 18 '13 at 7:49

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