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The authors are dealing with the Dirichlet integral-type biholomorphic-invariant pseudodistance ${\rho }_X^{\alpha }(z_0,z)\left[\right]$ introduced by Dolbeault and Ławrynowicz (1989) in connection with bordered holomorphic chains of dimension one. Several properties of the related hyperbolic-like manifolds are considered remarking the analogies with and differences from the familiar hyperbolic and Stein manifolds. Likewise several examples are treated in detail.