Global behaviours of circles in a complex hyperbolic space
A unit speed curve \(\gamma: \mathbb{R}\to M\) in a Riemannian manifold \(M\) is called a circle iff there exists \(k\in \mathbb{R}_+\) and a unit vector field \(Y\) along the curve such that \(\nabla_\partial \dot\gamma =kY\) and \(\nabla_\partial Y= -k\dot \gamma\), where \(\partial\) denotes the canonical unit vector field of \(\mathbb{R}\). The behaviour of circles in real space forms is well-known. After \textit{T. Adachi}, \textit{S. Maeda} and \textit{S. Udagawa} have investigated circles in the complex projective space [Osaka J. Math. 32, 709-719 (1995; Zbl 0857.53034)], the present authors treat now the problem in the \(n\)-dimensional complex hyperbolic space \(\mathbb{C} H^n\). Here the circles are characterized (up to holomorphic isometry) by their curvature \(k \) and their ``complex torsion'' \(\tau: =\langle \dot\gamma, JY\rangle \in[-1,1]\). By means of these entities the authors can characterize bounded and unbounded circles, in the case of bounded circles the periodic ones and in the case of unbounded ones their behaviour at \(\pm\infty\).
scientific article; zbMATH DE number 1102908