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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
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