Kaeler magnetic fields and graphs

点击翻译按钮获取中文摘要

In order to study properties of submanifolds in Kaehler manifolds, we consider natural closed 2-forms on submanifolds which are induced by almost contact metric structures. We study motions of electric charged particles of unit speed under actions of Sasakian magnetic fields, which are constant multiples of natural 2forms, on submanifolds. Since strengths of Sasakian magnetic fields acting on charged particles are not uniform, we are interested in the problem whether there exist trajectories which are also curves of order 2. From the viewpoint of classical Frenet-Serre formula, we may say that curves of order 2 are simple. We study submanifolds in complex space forms, which are highly symmetric. We show that there exist such trajectories on real hypersurfaces of type A whose class includs geodesic spheres. We can characterize these real hypersurfaces by the amount of such trajectories. We also show how lengths of closed trajectories which are also curves of order 2 are distributed on the real line.Besides, we give discrete models of Kaehler manifolds. We propose graphs whose edges are colored by 2 colors to be good models. We studied distribution of closed passes on these graphs and show a similarity between the distribution of lengths of closed passes on graphs and that of closed trajectories on Kaehler manifolds.