The symmetrized bidisc \[ G \stackrel{ \rm{ def}}{ =}\{ (z+w,zw):|z|<1,\ |w|<1\}, \] under the Carath\'eodory metric, is a complex Finsler space of cohomogeneity $1$ in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, $G$ does not admit a natural notion of angle, but we nevertheless show that there { \em is} a notion of orthogonality. The complex tangent bundle $TG$ splits naturally into the direct sum of two line bundles, which we call the { \em sharp} and { \em flat} bundles, and which are geometrically defined and therefore covariant under automorphisms of $G$. Through every point of $G$ there is a unique complex geodesic of $G$ in the flat direction, having the form \[ F^\beta \stackrel{ \rm{ def}}{ =}\{ (\beta+\bar\beta z,z)\ : z\in\mathbb{ D}\} \] for some $\beta \in\mathbb{ D}$, and called a { \em flat geodesic}. We say that a complex geodesic \emph{ $D$ is orthogonal} to a flat geodesic $F$ if $D$ meets $F$ at a point $\lambda$ and the complex tangent space $T_\lambda D$ at $\lambda$ is in the sharp direction at $\lambda$. We prove that a geodesic $D$ has the closest point property with respect to a flat geodesic $F$ if and only if $D$ is orthogonal to $F$ in the above sense. Moreover, $G$ is foliated by the geodesics in $G$ that are orthogonal to a fixed flat geodesic $F$.
Categories: math.DG math.CV