# Schwarz–Pick Theorem for Higher-dimensional Hyperbolic Space?

According to Wikipedia, Schwarz–Pick theorem says that that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric, which also means that it decreases distances of points for a two-dimensional hyperbolic space under Poincaré disk model.

My question is: Is there an analog of this theorem for three or higher dimensional hyperbolic space, probably under higher-dimensional Poincaré disk model?

• Comments are not for extended discussion; this conversation has been moved to chat. Jun 4 '21 at 8:51

There exists an analogue for holomorphic map from the unit ball in $$\bf C^n$$ to itself, or even every bounded open set in $$\bf C^n$$.
In fact every complex manifold $$M$$ (for instance open set in $$C^n$$) such that every map $$\bf C\to M$$ is constant (for instance bonded open set) admit a natural distance called the Kobayashi distance which is decreased by holomorphic maps.
The Kobayashi distance of the ball in $$\bf C^n$$ is the natural generalization of the Poincaré metric :its isometry group is $$SU(n,1)$$.