Definition S is isometric to the plane $\mathbb{R^2}$

Question

The parameterized surface $S\subseteq \mathbb{R^3}$ given by
$$(u,v)\mapsto(u,v,u^2)$$I want to show that $S$ is isometric to the flat plane $\mathbb{R^2}$.

What is the definition of "$S$ is isometric to something"?

Dose it mean $S$ is isometry?

Answer 1

To say that two metric spaces $X$ and $Y$ are isometric means that there is a homeomorphism $f \colon X \to Y$ that preserves distances. I.e. $$d_Y(f(a),f(b)) = d_X(a,b)$$ where $d_X$ and $d_Y$ are the distance functions on $X$ and on $Y$ and $a,b$ are points in $X$.

For example, if $X$ is the line $\{(x,x) \in \mathbb{R}^2 : x \in \mathbb{R}\}$ and $Y$ is the real line, then the map $f(x,x) = x$ is a homeomorphism, but not and isometry because $d_X((a,a),(b,b)) = \sqrt2d_Y(a,b)$. So the isometric map is $g(x,x) = \sqrt2 x$.

The phrase "$S$ is an isometry" is nonsensical: isometries are functions, not surfaces.