This brief note was inspired by this MathOverflow post. A definition of a Kobayashi hyperbolic manifold is available here. Informally, it says that one can’t holomorphically map arbitrarily large complex unit disks into the complex manifold. In a certain sense “most” complex manifolds are Kobayashi-hyperbolic. Below is the proof of the following theorem of Royden, proved in this paper.
Theorem 1 Suppose is a holomorphic fibration, with fiber a Kobayashi-hyperbolic manifold . Then the fibration is locally flat, i.e. the gluing maps on overlapping local charts must be (locally) constant.
Remark 1 The requirement that the fibration be holomorphic means that locally it is holomorphically isomorphic to . For example, if is the unit ball in , the natural projection to the unit disk in is not a holomorphically trivial fibration. One can see this using the maximum principle (see the MO post for details).