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Une remarque sur l’hyperbolicité injective. (A remark about injective hyperbolicity). (French) Zbl 0741.32019
A domain $$D$$ is called $$S$$-hyperbolic, if the injective Kobayashi pseudodistance defines the topology in $$D$$. This pseudodistance, introduced by Hahn, is in general greater than the usual Kobayashi pseudodistance. Hence each hyperbolic domain is $$S$$-hyperbolic. The opposite is not true. However, the author proves that if $$D_ 1$$ and $$D_ 2$$ are subdomains of normed complex vector spaces of positive dimension, then $$D_ 1\times D_ 2$$ is $$S$$-hyperbolic iff it is hyperbolic and iff both $$D_ i$$ are hyperbolic.
##### MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds