论文标题
内在方向，正交性和在对称的bidisc中的杰出的大地学
Intrinsic Directions, Orthogonality and Distinguished Geodesics in the Symmetrized Bidisc
论文作者
论文摘要
The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}, \] under the Carathéodory metric, is a complex Finsler space of cohomogeneity $1$ in which the geodesics, both real and complex, enjoy a rich geometry.作为Finsler歧管，$ G $不承认自然的角度概念，但是我们表明{\ em是}正交性的概念。复杂的切线束$ tg $自然地分成两个线束的直接总和，我们将其称为{\ em Sharp}和{\ em Flat}捆绑包，它们在几何定义上定义，因此在$ g $的自动形态下协变。在$ g $的每个点上，都有一个独特的复杂地理测量，$ g $在平坦的方向上具有\ [f^β\ stackrel {\ rm {frm {def}} {=} {=} {=} \ {（β+\barβz，z） $β\ in \ Mathbb {d} $，并称为{\ em flat Geodesic}。我们说，如果$λ$以$λ$，$ d $符合$ f $，则复杂的Geodesic \ emph {$ d $是正交}，与平坦的大地测量$ f $相遇，而复杂的切线太空$t_λd$ at $λ$在$λ$的方向上朝着尖锐的方向上。我们证明，在上述意义上，当$ d $正交到$ f $时，地球$ d $与平坦的大地测量$ f $相对于平坦的地球$ f $具有最接近的属性。此外，$ g $是由Geodesics以$ g $为叶的，与固定的平面地球$ F $正交。
The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}, \] 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 {\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 \[ F^β\stackrel{\rm{def}}{=}\{(β+\barβz,z)\ : z\in\mathbb{D}\} \] for some $β\in\mathbb{D}$, 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 $λ$ and the complex tangent space $T_λD$ at $λ$ is in the sharp direction at $λ$. 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 $F$.
