Integral geometry and variational problems in homogeneous spaces

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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.