Let $(X, d_0), (Y, d_1)$ be compact metric spaces. Consider the metric $d$ on $X\times Y$ defined by $d((x_0, y_0), (x_1, y_1)=\max \{d_0(x_0, x_1), d_1(y_0, y_1)\}$.
Assume that $F:X\times Y\to X\times Y$ is an isometry. Are there isomety maps $f:X\to X$ and $g:Y\to Y$ such that $F(x, y)= (f(x), g(y))$?
For this, fix $a\in X, b\in Y$. We have $d(F(a, y), F(a, y'))= d_1(y, y')$ and $d(F(x, b), F(x', b))= d_0(x, x')$.
Can I say if $f(x)= \pi_1(F(x, b))$ and $g(y)= \pi_2(F(a, y))$, then $f:X\to X$ and $g:Y\to Y$ are isometry?
Not in general. For example, a rotation in $\mathbb R^2$ through $\frac\pi2$ is an isometry with respect to the maximum metric, but in this case the functions $f$ and $g$ are constant and thus far from isometries.