Proper quasi-homogeneous domains in flag manifolds and geometric structures

TL;DR

This paper investigates the existence and properties of proper quasi-homogeneous domains in flag manifolds, revealing non-existence in many cases and establishing convexity and metric properties where they do exist.

Contribution

It demonstrates the non-existence of such domains in many flag manifolds and characterizes the convexity and metric structure of those that do exist.

Findings

No proper quasi-homogeneous domains in many flag manifolds.

Existence of convexity and invariant metrics in certain cases.

Restrictions on developing maps for specific $(G,X)$-structures.

Abstract

In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we show that they satisfy a natural convexity condition and have an invariant metric which generalizes the Hilbert metric. As an application we give some restrictions on the developing map for certain $(G,X)$-structures.