  We first give an exposition on holomorphic isometries from the Poincar\'e
disk to polydisks and from the Poincar\'e disk to the product of the Poincar\'e
disk with a complex unit ball. As an application, we provide an example of
proper holomorphic map from the unit disk to the complex unit ball that is
irrational, algebraic and holomorphic on a neighborhood of the closed unit
disk. We also include some new results on holomorphic isometries.
