Intrinsic Directions, Orthogonality and Distinguished Geodesics in the Symmetrized Bidisc

TL;DR

This paper explores the rich geometric structure of the symmetrized bidisc under the Carathéodory metric, introducing notions of orthogonality and geodesic foliation based on intrinsic directions and splitting of the tangent bundle.

Contribution

It defines a covariant splitting of the tangent bundle into sharp and flat bundles, introduces a notion of orthogonality for geodesics, and characterizes geodesics with the closest point property relative to flat geodesics.

Findings

Existence of a natural orthogonality notion in the symmetrized bidisc.

Unique flat geodesics through each point of the domain.

Foliation of the domain by geodesics orthogonal to a fixed flat geodesic.

Abstract

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 $F$.