Bounded Circles on a Complex Hyperbolic Space are Expressed by Trajectories on Geodesic Spheres

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (1) 2024, pp. 1–10
(Received: 2023-12-06 , Revised: 2023-12-27)

Yusei Aoki , Division of Mathematics and Mathematical Science, Nagoya Institute of Technology, Nagoya 466-8555, Japan; e-mail: yusei11291@outlook.jp

Toshiaki Adachi , Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan; e-mail: adachi@nitech.ac.jp

Abstract/Résumé:

We take a bounded circle on a complex hyperbolic space. We show that if it has complex torsion either \(\pm 1\) or \(0\) then it is expressed by a geodesic on some geodesic sphere, and show that if it has complex torsion \(\tau\) with \(0 < |\tau| < 1\) then it is uniquely expressed by a non-geodesic trajectory on a geodesic sphere up to congruency.

Nous prenous un cercle borné en l’espace hyperbolique complexe. Nous montrons que il est exprimé par une géodésique sur une sphère géodésique si sa torsion complexe est \(0\) ou \(\pm 1\), et montrons que il est uniquement exprimé par une trajectoire sur une sphère géodésique qui n’est pas une géodésique si sa torsion complexe est \(0 < |\tau| < 1\).

Keywords: Geodesic spheres, circles, complex torsions, congruent, extrinsic shapes

AMS Subject Classification: Hermitian and K_õhlerian structures, Sub-Riemannian geometry, Geodesics 53B35, 53C17, 53C22

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