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 1Suppose 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 1The requirement that the fibration be holomorphic means that locally it isholomorphicallyisomorphic to . For example, if is the unit ball in , the natural projection to the unit disk in isnota holomorphically trivial fibration. One can see this using the maximum principle (see the MO post for details).