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The authors are dealing with the Dirichlet integral-type biholomorphic-invariant pseudodistance ρX(z0, z)[U ] introduced by Dolbeault and Lawrynowicz (1989) in connection with bordered holomorphic chains of dimension one. Several properties of the related hyperboliclike manifolds are considered remarking the analogies with and differences from the familiar hyperbolic and Stein manifolds. Likewise several examples are treated in detail.

Hyperbolic-like manifolds, geometrical properties and holomorphic mappings