A Kobayashi pseudo-distance for holomorphic bracket generating distributions

Abstract:

We generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow’s theorem in sub-Riemannian geometry. Let $G$ be a linear semisimple Lie group. For a complex $G$-homogeneous manifold $M$ with a $G$-invariant holomorphic bracket generating distribution $D$, we prove that $(M,D)$ is Kobayashi hyperbolic if and only if the universal covering of $M$ is a canonical flag domain and the induced distribution is the superhorizontal distribution.