A geometric characterization of the symmetrized bidisc

The symmetrized bidisc

G 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 in G that is invariant under all automorphisms of G. Moreover, G is foliated by those complex geodesics that meet 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.