The symmetrized bidisc

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

for some \(\beta \in \mathbb {D}\), and called a flat geodesic. We say that a complex geodesic 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.

对称的比迪奇

在Carathéodory度量下，是一个复杂的Finsler同质性空间，其中实测和复杂的测地线都具有丰富的几何形状。作为Finsler流形，摹不承认角度的自然概念，但是我们仍然表明，是正交的概念。复杂的切线束TG自然地分裂成两个线束的直接总和，我们称其为尖束束和扁平束，它们在几何上是定义的，因此在G的自同构下是协变的。通过的每个点ģ，存在的唯一复杂测地ģ在扁平方向，具有该形式

对于一些\（\ beta \ in \ mathbb {D} \），并称为平面测地线。我们说一个复杂的测地d 是垂直于平坦的大地˚F如果d满足˚F在点\（\拉姆达\）和复杂的切空间\（T _ \拉姆达d \）在\（\拉姆达\）是在\（\ lambda \）处的方向清晰。我们证明，当且仅当D在上述意义上与F正交时，测地线D相对于平面测地线F具有最接近的点属性。而且，G由G中与固定平坦测地线F正交的测地线构成。