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

Abstract

AbstractThe symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z|<1,\quad |w|<1\}, \end{aligned}$$ G = def { ( z + w , z w ) : | z | < 1 , | w | < 1 } , under the Carathéodory 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 is a notion of orthogonality. The complex tangent bundle TG splits naturally into the direct sum of two line bundles, which we call the sharp and 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 $$\begin{aligned} F^\beta {\mathop {=}\limits ^\mathrm{{def}}}\{(\beta +{\bar{\beta }} z,z)\ : z\in \mathbb {D}\} \end{aligned}$$ F β = def { ( β + β ¯ z , z ) : z ∈ D } for some $$\beta \in \mathbb {D}$$ β ∈ D , and called a flat geodesic. We say that a complex geodesic Dis orthogonal to a flat geodesic F if D meets F at a point $$\lambda $$ λ and the complex tangent space $$T_\lambda D$$ T λ D at $$\lambda $$ λ is in the sharp direction at $$\lambda $$ λ . 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.