On the relative intrinsic pseudo distance and the hyperbolic imbeddability. (Q1396077)

The authors establish a relation between the Kobayashi relative intrinsic distance on holomorphic fiber bundles and one in the base space. Moreover, they prove the following: Suppose that \((\overline Z,\pi, Z)\) is a fiber bundle with compact hyperbolic fiber over a complex manifold \(Z\) and \(M\subset Z\) is a complex subspace with \(d_{M,Z}\) that induces the given topology on \(\overline M\). Then \(M\) is hyperbolically imbedded in \(Z\) if and only if \(\pi^{-1}(M)\) is hyperbolically imbedded in \(\overline Z\).