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In the following example, we want to show that a catenoid and a helicoid are locally isomoprhic by comparing the first fundamental form. (Example from Do Carmo, page 224, section 4.2, example 2).

Example $\quad$ Let $S$ be a surface of revolution and let \begin{equation} x(u,v) = (f(v)\cos{u}, f(v)\sin{u}, g(v)), \quad a < v < b, \quad 0 < u < 2\pi, \quad f(v) > 0 \end{equation} be a parametrization of $S$. The coefficients of the first fundamental form of $S$ in the parametrization $x$ are given by \begin{equation} E = (f(v))^2, \quad F = 0, \quad G = (f'(v))^2 + (g'(v))^2. \end{equation} In particular, the surface of revolution of the catenary, \begin{equation} x = a\cosh{v}, \quad z = av, \quad -\infty < v < \infty \end{equation} has the following parametrization \begin{equation} x(u,v) = (a\cosh{v}\cos{u}, a\cosh{v}\sin{u}, av), \quad 0 < u < 2\pi, \quad -\infty < v < \infty \end{equation} relative to which the coefficients of the first fundamental form are \begin{equation} E = a^2\cosh^2{v}, \quad F = 0, \quad G = a^2(1 + \sinh^2{v}) = a^2\cosh^2{v} \end{equation} The surface of revolution is called the catenoid. We shall show that the catenoid is locally isometric to the helicoid parametrized by \begin{equation} \bar{x}(\bar{u},\bar{v}) = (\bar{v}\cos{\bar{u}}, \bar{v}\sin{\bar{u}}, a\bar{u}), \quad 0 < \bar{u} < 2\pi, \quad -\infty < \bar{v} < \infty \end{equation} Let us make the following change of parameters \begin{equation} \bar{u} = u, \quad \bar{v} = a\sinh{v}, \quad 0 < u < 2\pi, -\infty < v < \infty \end{equation} So, I know that the latter change of parameters is possible since the map is injective and the jacobian is nonzero everywhere. Then, they go on to compute the first fundamental of the helicoid of the example, which is the same as the first fundamental form of the catenoid.

However, how did they come up with this change of parameters? It feels it bit like you can modified it to anything, as long as the map is injective and the jacobian is non-zero everywhere. Thus: how should I pick such a change of parameters? What is a proper strategy?

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    $\begingroup$ You wrote down the first fundamental form for the catenoid. Now write down the first fundamental form for the helicoid (in terms of $\bar u,\bar v$). How do you make the two fundamental forms "line up" (i.e., $E=\bar E$, $F=\bar F$, $G=\bar G$)? $\endgroup$ – Ted Shifrin Jun 10 at 20:30
  • $\begingroup$ Yes, I did that and found a satisfying result. However, I wanted to know whether there is a way without first computing the first fundamental form and then picking your paramaters such that it comes uit nicely $\endgroup$ – Rice4000 Jun 10 at 22:12
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    $\begingroup$ No. Think about what characterizes a local isometry in the first place. (You said "locally isomorphic," but you meant "locally isometric.") This means that you have a (local) diffeomorphism that pulls back the first fundamental form of one surface to the first fundamental form of the other. $\endgroup$ – Ted Shifrin Jun 10 at 22:19

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