An equiareal map that is not a local isometry

Question

Could you give an example of an equiareal map that is not a local isometry?

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A local diffeomorphism $f : S_1 \rightarrow S_2$ is equiareal if and only if, for any surface patch $\sigma (u, v)$ on $S_1$, the first fundamental forms $$E_1du^2 + 2F_1dudv + G_1dv^2 \ \text{ and } \ E_2du^2 + 2F_2dudv + G_2dv^2$$ of the patches $\sigma$ on $S_1$ and $f \circ \sigma$ on $S_2$ satisfy $$E_1G_1− F^2_1 = E_2G_2 − F^2_2$$

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A local diffeomorphism $f : S_1 \rightarrow S_2$ is a local isometry if and only if, for any surface patch $\sigma_1$ of $S_1$, the patches $\sigma_1$ and $f \circ \sigma_1$ of $S_1$ and $S_2$, respectively, have the same first fundamental form.

Answer 1

$$ (x,y) \mapsto (x+y,y) $$ in the ordinary plane

Answer 2

Hint: Let $S_{1} = S_{2} = \mathbf{R}^{2}$, and define $f(x, y) = (ax, by)$ for some positive real numbers $a$ and $b$. Under what conditions on $a$ and $b$ is $f$:

1. Area-preserving?

2. A local isometry?

If you'd like compact examples, it's straightforward to adapt this idea to maps between tori.