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.
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.