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

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Jim Agler , Zinaida A. Lykova , N. J. Young · YOU? · · 2020 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2012.03304

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 $F$.

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preprint

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en

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http://arxiv.org/abs/2012.03304

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https://doi.org/10.48550/arxiv.2012.03304

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Intrinsic Directions, Orthogonality and Distinguished Geodesics in the Symmetrized Bidisc

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preprint

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en

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2020

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2020-12-06

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Jim Agler, Zinaida A. Lykova, N. J. Young

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https://arxiv.org/abs/2012.03304

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Geodesic, Mathematics, Orthogonality, Tangent space, Tangent bundle, Pure mathematics, Lambda, Combinatorics, Tangent vector, Space (punctuation), Manifold (fluid mechanics), Mathematical analysis, Geometry, Tangent, Physics, Engineering, Linguistics, Mechanical engineering, Philosophy, Optics

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