Integral geometry and variational problems in homogeneous spaces

齐次空间中的积分几何和变分问题

基本信息

批准号：
16540051
负责人：
TASAKI Hiroyuki
金额：
$ 2.18万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

项目摘要

In the last academic year we established a Crofton formula in Riemannian symmetric spaces by the use of reflective submaifolds, which is totally geodesic. In order to get explicite expression of the Crofton formula we need some geometric invariants of submanifolds. In the case where the head investigator gave an explicit expression of Poincare formula of submanifolds in complex space forms, he introduced a notion of multiple Kahler angle. By the use of the multiple Kahler angle he could obtain an explicit Crofton formula of submanifolds in complex space forms. For the purpose that we extend the class of submanifolds we use in Crofton formula in Riemannian symmetric spaces, we extend the notion of reflective submanifolds to that of weakly reflective submanifolds. Weakly reflective submanifolds are special ones of minimal submanifolds. Some arguments show that austere submanifolds are weakly reflective. In order to get austere submanifolds in the spheres we described a condition for a submanifold to be austere among the orbits of the linear isotropy actions of Riemannian symmetric pairs. As a result of this we obtained a classification of austere orbits in the spheres under the linear isotropy actions. We observed that some of them are invariant under an isometry of the sphere which reverses the submanifold with respect to the normal directions. So we called such submanifolds weakly reflective submanifolds and started our research of weakly reflective submanifolds.

在最后一学年，我们通过使用反射式submaifolds建立了Riemannian对称空间中的Crofton公式，这完全是大地的。为了获得Crofton公式的显式表达，我们需要一些子手机的几何不变。如果主管调查员以复杂的空间形式对Submanifolds的Poincare公式表示明确表达，则他引入了多个Kahler角度的概念。通过使用多个Kahler角度，他可以在复杂的空间形式中获得Submanifolds的明确Crofton公式。为了将我们在Riemannian对称空间中使用的Submanifold类别扩展到crofton公式中，我们将反射性子策略的概念扩展到弱反射性的子手势的概念。弱反射性的子手势是最小的亚曼叶夫的特殊submanifolds。一些论点表明，严峻的子曼福尔德反射较弱。为了在球体中获得严峻的子延伸，我们描述了子曼群在riemannian对称对的线性各向同性动作的轨道中均匀的条件。结果，我们在线性各向同性作用下获得了球体中的严重轨道的分类。我们观察到，其中一些是在球体的等轴测图下不变的，该球体相对于正常方向逆转了子序列。因此，我们称这样的submanifolds弱反思性的子曼属群，并开始研究弱反思性的子曼叶。