Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc

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