Kaeler magnetic fields and Carnot spaces

凯勒磁场和卡诺空间

基本信息

批准号：
11640073
负责人：
ADACHI Toshiaki
金额：
$ 2.24万
依托单位：
Nagoya Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640073/
关键词：
Kaehler magnetic field length spectrum geodesic sphere circle complex space form shperical mean helix Kaehler immersion Kaehler immersion Veronese emmbeding curve of order 2 kahler magnetic field closed geodesic geodesis sphere

项目摘要

The head investigator studied trajectories for Kaehler magnetic fields on symmetric spaces. His works, a part of which is a joint work with some coinvestigators, can be classified into the following four directions.(1) Mean operators associated with magnetic fieldsIn order to study the relationship between trajectories for Kaehler magnetic fields and Schroedinger operators, the head investigator studied magnetic random walks. On complex space forms the mean operator generated by this magnetic random walk has the same properties as that generated by the geodesic random walk. On the other hand, if we studied a magnetic spherical mean which is derived from potential on unit tangent bundle we found the principal term of the formal expansion was the Schroedinger operator.(2) Circles on complex space formsExtending the notion of trajectories for Kaehler magnetic fields the head investigator studied length spectrum for circles on complex space forms. The moduli space of circles on these space … More s are open rectangles in a Euclidean plane parametrized by geodesic curvature and complex torsion. Concerning the continuity of length spectrum for circles we found it had a natural foliation structure. By use of this structure we clearfied set theoretic properties of length spectrum, the asaymptotic behavior of the number of congruency classes of circles with respect to their length, and the properties of the k-th length spectrum function with respect to the geodesic curvature.(3) Geodesics on geodesic spheres in a rank one symmetric spacesHaving been inspired with the idea in our study on circles we studied lengths of geodesics on a geodesic sphere in a rank one symmetic space, which is famous as an example of Berger sphere. We considered geodesics on a geodesic sphere as curves on a complex space form, and studied their horizontal lifts with respect to the Hopf fibration. We could then treat them as curves in a Euclidean space. We showed the relationship between the radius of a geodesic sphere and length-simplicity of geodesics and clearfied the asymptotic behavior of the number of closed geodesics on a geodesic sphere with respect to their length.(4) Characterizations of submanifolds in complex space formsBy use of properties of circles and helices on complex space forms the head investigator and S. Maeda characterized submanifolds in complex space forms. Their idea stands on the technique of treating geodesic and circles on a submanifold as curves on a complex space form. They characterized homogeneous submanifold, Veronese embeddings and some other important submanifolds. Less

首席研究员研究了对称空间上凯勒磁场的轨迹，其中一部分是与一些共同研究人员的共同工作，可分为以下四个方向：（1）与磁场相关的均值算子。凯勒磁算子和薛定谔算子的轨迹之间的关系，首席研究员研究了磁随机场游走，在复杂空间形式上，由这种磁随机游走生成的均值算子具有与测地随机游走生成的相同的属性。另一方面，如果我们研究从单位切丛上的势导出的磁球平均值，我们发现形式展开的主要项是薛定谔算子。（2）复杂空间形式上的圆扩展了凯勒磁场磁头轨迹的概念研究人员研究了复杂空间形式上的圆的长度谱，这些空间上的圆的模空间是欧几里得平面中的开放矩形，由测地曲率和复数参数化。关于圆的长度谱的连续性，我们发现它具有自然的叶状结构，通过使用这种结构，我们阐明了长度谱的集合理论属性，即圆的同余类数量相对于其长度的渐近行为，以及第k个长度谱函数相对于测地曲率的性质。(3)一阶对称空间中测地球上的测地线受到我们研究中的想法的启发在圆上，我们研究了一级对称空间中测地球上的测地线长度，这是著名的伯杰球的例子。我们将测地球上的测地线视为复杂空间形式上的曲线，并研究了它们的水平升力。然后我们可以将它们视为欧几里得空间中的曲线，我们展示了测地球的半径与测地线的长度简单性之间的关系，并阐明了测地线球上闭合测地线的数量相对于其长度的渐近行为。 (4) 复空间形式中子流形的表征通过使用复空间形式上的圆和螺旋的性质，首席研究员和 S. Maeda 表征了复空间形式中的子流形他们的想法基于将子流形上的测地线和圆视为复杂空间形式上的曲线的技术，他们描述了齐次子流形，Veronese。嵌入和其他一些重要的子流形。