Kaeler magnetic fields and Carnot spaces

凯勒磁场和卡诺空间

• 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

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。他的著作主要有以下四个方面，其中一部分是与其他研究者的合作。(1)为了研究Kaehler磁场和Schroedinger算子的轨迹之间的关系，首席研究员研究了磁随机游动。在复空间形式上，这种磁随机游动产生的平均算子具有与测地线随机游动产生的平均算子相同的性质。另一方面，如果我们研究由单位切丛上的势导出的磁球平均，我们发现形式展开式的主项是Schroedinger算子。(2)复杂空间形式上的圆扩展凯勒磁场轨迹的概念，首席研究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 ...更多信息 s是欧氏平面中的开矩形，由测地曲率和复挠率参数化。关于圆的长度谱的连续性，我们发现它具有自然的叶理结构。利用这种结构，我们阐明了长度谱的集合论性质，圆的同余类数关于其长度的渐近性态，以及第k阶长度谱函数关于测地曲率的性质. (3)受圆的研究思想的启发，我们研究了一阶对称空间中测地球上测地线的长度，这是著名的Berger球的例子。我们把测地球面上的测地线看作是复空间形式上的曲线，并研究了它们相对于Hopf纤维化的水平提升。然后我们可以把它们当作欧几里得空间中的曲线。给出了测地球半径与测地线的长度简单性之间的关系，明确了测地球上闭测地线个数关于其长度的渐近性态。(4)复空间形式中子流形的刻划利用复空间形式中圆和螺旋的性质，本文的主要研究者和S。Maeda刻画了复空间形式中的子流形。他们的想法站在技术的处理测地线和圆的子流形上的曲线的复杂空间形式。他们刻画了齐次子流形、Veronese嵌入等重要子流形。少

项目成果

Toshiaki ADACHI: "Length spectrum of circles and Kaehler magnetic fields on complex space forms"Aspects of complex analysis, differential geometry, mathematical physics, and applications, World Scientific. 172-182 (1999)

Toshiaki ADACHI：“复杂空间形式上的圆长度谱和凯勒磁场”复分析、微分几何、数学物理和应用方面，世界科学。

Toshiaki ADACHI: "Length spectrum of geodesic sptieves in a non-flat complex space form"Journal of Mathematical Sociaty of Japan.

Toshiaki ADACHI：“非平坦复空间形式中的测地线长度谱”日本数学会杂志。

Sadahiro MAEDA: "Geometric meaning of isoparametric hypersurfaces in a real space form"Canadian Mathematical Bulletin. 43. 74-78 (2000)

Sadahiro MAEDA：“实空间形式中等参超曲面的几何意义”加拿大数学公报。

Toshiaki ADACHI: "Some charaterizations of quaternionic space forms"Proceedings of JAPAN Acadey of Sciences, Series A. 76・10. 168-172 (2000)

Toshiaki ADACHI：“四元空间形式的一些特征”，日本科学院院刊，系列 A. 76・10（2000）。

Toshiaki ADACHI: "Some aspects on circles and helices in a complex projective space"Memoirs of the Faculty of Science and Engineering, Shimane University. 32. 1-8 (1999)

安达俊明：《复杂射影空间中圆和螺旋的某些方面》岛根大学理工学院回忆录。