For a domain \(D \subset \mathbb C^n\) the Carathéodory distance function is denoted by \(c_D\), and the Lempert function is defined by \[ \widetilde k_D(w,z)=\inf \{ p(\lambda_1,\lambda_2) : \text{there is } f \in {\mathcal O}(\mathbf{D}, D) \text{ with } f(\lambda_1)=w,\;f(\lambda_2)=z\}. \] Here \(p\) is the Poincaré distance on the unit disc \(\mathbf{D}\) in the plane.
Given two points \(w,z\in D\), \(w\neq z\), a \(\widetilde k_D\)-geodesic for \((w,z)\) is a holomorphic mapping \(\varphi: \mathbf{D}\rightarrow D\), such that, for suitable \(\lambda_1,\lambda_2 \in \mathbf{D}\) one has \(\varphi (\lambda_1)= w\), \(\varphi(\lambda_2) =z\) and \(p(\lambda_1, \lambda_2)= \widetilde k_D(w,z)\). A holomorphic mapping \(\varphi: \mathbf{D}\rightarrow D\) is called a complex geodesic, if \(c_D(\varphi (\lambda_1), \varphi (\lambda_2))= p(\lambda_1,\lambda_2)\), for any \(\lambda_1,\lambda_2 \in \mathbf{D}\). For any complex geodesic \(\varphi\) one has \[ c_D(\varphi (\lambda_1), \varphi (\lambda_2))= \widetilde k_D( \varphi (\lambda_1), \varphi (\lambda_2)). \] On the bidisc \(\mathbf{D}^2\) in \(\mathbb C^2\) we define the mapping \[ \pi (\lambda_1, \lambda_2):= (\lambda_1+\lambda_2, \lambda_1\lambda_2). \] The domain \(G_2 = \pi (\mathbf{D}^2 )\) is then called the symmetrized bidisc. We let \(S\) denote the image of the diagonal of the bidisc under \(\pi\), explicitly \(S:=\{(2\lambda , \lambda^2) :\lambda \in \mathbf{D}\}\).
The aim of the present article is a description of the complex geodesics in \(G_2\). In this domain one even has \(c_{G_2}= \widetilde k_{G_2}\). The main result of the paper now is the following: A holomorphic mapping \(\varphi: \mathbf{D}\rightarrow D\) is a complex geodesic, if and only if one of the following two conditions are fulfilled (i) If \(\varphi (\mathbf{D}) \cap S \neq \emptyset\), then \(\varphi= \widetilde \varphi \circ b_\alpha \), where \(\alpha \in \mathbf{D},\, \widetilde \varphi (\lambda) = \pi (B(\sqrt{\lambda} ), B(-\sqrt{\lambda}))\), \(B\) is a Blaschke product of degree one or two, and \(b_\alpha (\lambda) = \frac{\lambda-\alpha}{1-\overline \alpha \lambda}\). (ii) If \(\varphi (\mathbf{D}) \cap S = \emptyset\), then \(\varphi= \pi \circ f\), where \(f_1,f_2\) are automorphisms of the unit disc such that \(f_1-f_2\) has no root in the unit disc.
As a corollary the authors derive: For a complex geodesic \(\varphi\) in \(G_2\) one of the three cases can occur (i) \(\varphi (\mathbf{D}) \cap S = \emptyset\) (ii) \(\varphi (\mathbf{D}) \cap S \) consists of exactly one point (iii) \(\varphi (\mathbf{D}) =S \). Moreover, each of the three possibilities (i)–(iii) is satisfied by some complex geodesic.