Integral geometry and variational problems in homogeneous spaces

在上一学年，我们利用完全测地的反射子流形，在黎曼对称空间中建立了一个Crofton公式。为了得到Crofton公式的显式表示，我们需要一些子流形的几何不变量。在首席研究员给出了复空间形式下子流形的Poincare公式的显式表示的情况下，他引入了多重Kahler角的概念。利用多重Kahler角，他可以得到复空间形式下子流形的显式Crofton公式。为了在黎曼对称空间中推广Crofton公式中使用的子流形，我们将反射子流形的概念推广到弱反射子流形的概念。弱反射子流形是极小子流形的一种特殊形式。一些论证表明，简约子流形是弱反射的。为了得到球面上的素子流形，我们刻画了黎曼对称对的线性各向同性作用的轨道中的子流形为素子流形的条件。作为结果，我们得到了在线性各向同性作用下球体中的硬轨道的分类。我们观察到，它们中的一些在球面的等距下是不变的，该球面相对于法线方向反转子流形。因此，我们称这种子流形为弱反射子流形，并开始了对弱反射子流形的研究。

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.