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Suppose $H$ is a finite dimensional reproducing kernel Hilbert space offunctions on $X.$ If $H$ has the complete Pick property then there is anisometric map, $\Phi,$ from $X,$ with the metric induced by $H,$ into complexhyperbolic space, $\mathbb{CH}^{n},$ with its pseudohyperbolic metric. Weinvestigate the relationships between the geometry of $\Phi(X)$ and thefunction theory of $H$ and its multiplier algebra.

Complex Hyperbolic Geometry and Hilbert Spaces with the Complete Pick Property