Integral geometry in homogeneous spaces and its applications

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In the research of integral geometry in the homogeneous space, integration on the orbit of a linear isotoropy action performs the key role. Many of important examples are austere submanifolds. The representative, Ikawa and Sakai classified the austere orbits of linear isotoropy actions of the Riemann symmetry pair in the sphere. When those austere orbits were examined in detail, a lot of austere orbits have not only the symmetry of the second fundamental form, but a certain kind of global symmetry. Because this global symmetry was a character to weaken the definition of reflective submanifold, it was named weakly reflective submanifold and examined its basic property. Weakly reflective submanifolds are austere submanifolds, and austere submanifolds are minimal submanifolds. The weakly reflective orbits was able to be classified in addition to the classification of the above-mentioned austere orbits. It is shown that the orbit with degenerate Gauss map becomes a weakly reflective subman … More ifold, and has generalized this though there was a result that the orbit of cohomogeneity 1 with degenerate Gauss map becomes austere before. That is, orbits with degenerate Gauss map are weakly reflective, and weakly reflective orbits are austere.The research of the orbits of linear isotoropy action of Riemann symmetry pairs is important to lead various relations concerning kinematic formula and the quermassintegrals in real space forms and complex space forms. Actually, the concept of the multiple Kaehler angle that the representative introduced was able to be led from the viewpoint of geometry of the orbit naturally, and when kinematic formula in complex space forms was formulated, the character to have obtained from geometrical consideration in the orbit played a basic role. In addition, a multiple Kaehler angle and its basic properties are important bases in the re-construction of integral geometry that makes the concept of valuation that has progressed in the past, several years a base. The viewpoint of geometry of the orbit is indispensable to research integral geometry in this direction of the future. Less