# A Geometric Characterization of the Symmetrized Bidisc

## TL;DR

This paper provides a geometric characterization of the symmetrized bidisc, highlighting its unique properties related to complex geodesics and automorphisms, and establishing conditions that uniquely identify it among complex manifolds.

## Contribution

It introduces a set of geometric conditions that uniquely characterize the symmetrized bidisc within the class of complex manifolds.

## Key findings

- Existence of a unique invariant complex geodesic under all automorphisms.
- Foliation of the bidisc by specific complex geodesics meeting the invariant geodesic.
- Characterization of the bidisc through geometric properties and automorphism actions.

## Abstract

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\} \] has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic $\mathcal{R}$ in $G$ that is invariant under all automorphisms of $G$. Moreover, $G$ is foliated by those complex geodesics that meet $\mathcal{R}$ in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of $G$, characterize the symmetrized bidisc in the class of complex manifolds.