Let
\[
S^{2}=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}=1\right\},
\]
and let $d: S^{2} \times S^{2} \rightarrow \mathbb{R}$ be the restriction of the Euclidean metric on $\mathbb{R}^{3}$. If $f: S^{2} \rightarrow S^{2}$ is a map such that $d(f(x), f(y))=d(x, y)$ for all $x, y \in S^{2}$, then $f$ is surjective.
