# A Kobayashi pseudo-distance for holomorphic bracket generating distributions

## TL;DR

This paper extends the Kobayashi pseudo-distance to complex manifolds with holomorphic bracket generating distributions, linking hyperbolicity to the structure of universal coverings and flag domains.

## Contribution

It introduces a generalization of the Kobayashi pseudo-distance for such manifolds and characterizes hyperbolicity in terms of universal coverings and flag domains.

## Key findings

- (M,D) is Kobayashi hyperbolic iff the universal cover is a canonical flag domain.
- The induced distribution on the universal cover is the superhorizontal distribution.
- Provides a criterion for hyperbolicity based on geometric and algebraic structures.

## Abstract

In this paper, 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.