Schwarz-Pick Lemma

If $f$ is an analytic map of the Disk $\Bbb{D}$ into $\Bbb{D}$ and $f$ preserves the hyperbolic distance between any two points, then $f$ is a disk map and preserves all distance.

References

Busemann, H. The Geometry of Geodesics. New York: Academic Press, p. 41, 1955.