Study on submanfolds of homogeneous spaces from the view of orbital Grassmann geometry

从轨道格拉斯曼几何角度研究齐次空间的子流形

基本信息

批准号：
17540081
负责人：
NAITOH Hiroo
金额：
$ 1.96万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540081/
关键词：
Grassmann geometry homogeneous space submanifold Liegroup left invariant metric surface unimndnlar Lie noun リー郡

项目摘要

This study is on the Grassmann geometry on the Riemannian homogeneous spaces. Our aim is to consider the classification problem of extrinsic homogeneous submanifolds of Riemannian symmetric spaces. For this, in this study, we examine the case where a Riemannian homogeneous space is a 3-dimensional unimodular Lie group with a left invariant metric. The 3-dimensional unimodular Lie groups are classified into six ones; the 3-dimensional vector group, the 3-dimensional Heisenberg group, the groups of rigid motions of the Eucliden 2-plane and the Minkowski 2-plane, the special unitary group SU (2), and the special linear group SL (2,R). Also, for each of them the geometric properties such as the curvatures, the isometry group, and m on, can be expressed concretely. In this study we in particular consider the Grassmann geometry on the spaces SU (2) and SL (2,R), while the cases of the Heisenberg group and the groups of rigid motions of the Eucliden 2-plane and the Minkowski 2-plane are clarify by H Naitoh, J. Inoguchi, and K Kuwabara. The obtained main results are the following.;for both the spaces SU (2) and SL (2,R),(1) the classification for all the orbits associated with Grassmann geometries on their spaces(2) the determination of the orbits whose Grassmann geometries are nonempty(3) the analysis on the surface theory for nonempty Grassmann geometries, in particular, (3-1) the settlement of the existence problem of totally geodesic surfaces, (3-2) the settlement of the existence problem of flat surfaces, (3-3) the settlement of the existence problem of minimal surface, (3-4) the settlement of the existence problem of constant mean curvature surfaces(4) the overview of Grassmann geometry on all the 3-dimensional unimodular Lie groups with left invariant metrics.These results will be appeared in forthcoming papers titled by "Grassmann geometry on the 3-dimensional unimodular Lie groups I, II".

这项研究是关于里曼尼亚同质空间的格拉曼几何几何形状的。我们的目的是考虑Riemannian对称空间的外在均质次曼群的分类问题。为此，在这项研究中，我们研究了riemannian同质空间是一个三维的单型谎言群，左不变度度量。 3维单型谎言组分为六个；三维矢量组，三维的海森堡组，欧几里德2平面的刚性运动组和Minkowski 2平面，特殊的统一组SU（2）以及特殊线性组SL（2，2， R）。同样，对于每一个，都可以具体地表达诸如曲率，等轴测组和m之类的几何特性。在这项研究中，我们特别考虑了SU（2）和SL（2，R）上的Grassmann几何形状，而Heisenberg组的病例以及Eucliden 2 Plane和Minkowski 2 Plane的刚性运动组H Naitoh，J。Inoguchi和K Kuwabara澄清。获得的主要结果如下。；对于SU（2）和SL（2，R）的空间，（1）与Grassmann几何相关的所有轨道的分类（2）确定其轨道的分类格拉曼（Grassmann）的几何形状是非空的（3）对非空的格拉曼（Grassmann）几何形状的表面理论的分析，尤其是（3-1）存在完全测量表面的存在问题，（3-2）存在平坦的存在问题。表面，（3-3）解决最小表面的存在问题，（3-4）存在恒定平均曲率表面的存在问题（4）所有3维单模型层组上Grassmann几何学的概述使用左不变指标。这些结果将出现在即将发表的论文中，标题为“三维单型谎言组I，II，II”。