In "HOLOMORPHIC FIBER BUNDLES WITH HYPERBOLIC FIBER 1" by H. L. ROYDEN, it was proved that a holomorphic fiber bundle with simply connected base
and Kobayashi hyperbolic fiber is trivial. Now given a complex Lie group $G$ and a holomorphic $G$-principle fiber bundle $E$ over a simply connected complex manifold, may one applies the above theorem in this case? Namely, whether a complex Lie group $G$ is Kobayashi hyperbolic. If not in general, is it Kobayashi hyperbolic for $G=GL\_n(\mathbb{C})$ or $G=PGL\_n(\mathbb{C})$?



Kobayashi hyperbolic complex Lie groups
A complex Lie group is never Kobayashi hyperbolic, as the exponential map applied to any complex line in the Lie algebra gives a complex line in the group.


