Almost Complex Manifolds and Hyperbolicity

One of the sufficient conditions for a complex manifold to be (complete) hyperbolic (in the sense that its intrinsic pseudo-distance is a (complete) distance) is that it admits a (complete) Hermitian metric with holomorphic sectional curvature bounded above by a negative constant. The concept of hyperbolicity can be readily extended to almost complex manifolds. We will show that the above result for hyperbolicity can be generalized to the almost complex case. As an application, we prove that every point of an almost complex manifold has a complete hyperbolic neighborhood. In real dimension 4, this fact was established by Debalme and Ivashkovich 2] by a completely different method.