Abstract
The Kobayashi metric of a complex manifold dominates its Caratheodory metric. It thus seems natural to investigate cases of equality of these metrics, especially so since they correspond to the extremal case in Schwarz's lemma. The open unit polydisc A" in ~n is one of the simplest cases in which these metrics coincide, and the following theorem shows how these metrics characterize the complex structure of the polydisc. (Definitions of the metrics and the terms "finitely compact" and "indicatrix" appear in Sect. 2.)
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