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).
Proof: To prove the theorem, it suffices to check that any holomorphically varying isomorphism must be constant (here is the varying parameter). This will imply that the gluing maps are constant.
Next, it suffices to check this for small disks , so consider a holomorphic map , such that is a biholomorphism. By applying this same biholomorphism to both sides, we can assume , so it suffices to check is constant, for every (knowing ). Finally, observe that we can define by , with , in other words iterating .
Two facts will finish the proof. The first is a property of hyperbolic manifolds: for a local chart around and any natural , there exists a constant such that for any holomorphic map . The second is that the map has growing (in ) derivative.
Indeed, suppose in local coordinates
Then in local coordinates, we have
Here, the index is the first non-zero term in the -expansion. This implies that the -derivative vanishes to all orders, thus the map must be constant.