I will first overview the classical holomorphic isometry problem between complex manifolds, in particular between bounded symmetric domains. When the source is the unit ball, in general the characterization of holomorphic isometries to bounded symmetric domains is not quite clear. In a joint work with Shan Tai Chan, we recently characterize the holomorphic isometries from the unit disc to the product of the unit disc with the unit ball and obtained some interesting applications.

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