Kaeler magnetic fields and graphs

为了研究Kaehler流形上子流形的性质，我们考虑了由几乎接触度量结构诱导的子流形上的自然闭2-形式。研究单位速度带电粒子在Sasakian磁场作用下在子流形上的运动，Sasakian磁场是自然2形式的常数倍。由于作用在带电粒子上的Sasakian磁场强度不均匀，我们感兴趣的问题是是否存在也是二阶曲线的轨迹。从经典Frenet-Serre公式的观点来看，我们可以说二阶曲线是简单的。我们研究了复空间形式中的高度对称的子流形。我们证明了在类包含测地球面的A型实超曲面上存在这样的轨迹。我们可以通过这些轨迹的数量来描述这些真实的超曲面。给出了闭轨迹的长度在实直线上的分布，并给出了Kaehler流形的离散模型。我们建议边用两种颜色着色的图是很好的模型。我们研究了这些图上的闭路的分布，证明了图上的闭路的长度分布与Kaehler流形上的闭迹的长度分布相似。

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.