Differential geometry and integral geometry in homogeneous spaces and its applications

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We proved that the intersection of two real forms in a Hermitian symmetric space of compact type is an antipodal set in a joint work with Makiko Tanaka and investigated their intersection by the use of the theory of polars introduced by Chen-Nagano. Using this result we determined the Floer homology of two real forms and extended Arnold-Givental inequalities in a joint work with Hiroshi Iriyeh and Takashi Sakai. Moreover we used these results and a kinematic formula to obtain an estimate of the volume of Hamiltonian deformation of a real form from below.