Homogeneous spaces and variational problems

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The head investigator introduced the notion "multiple Kahler angle" and showed that we can describe integral geometry of submanifolds in complex projective spaces explicitly by the use of multiple Kahler angle. In the case of the complex projective plane he obtained with Kang more detailed Poincare formula. These Poincare formulae has an application on estimate of the area and the integral of Kahler angle of real surfaces. By this estimate we can get a minimizing solution of a certain variational problem. Moreover the head investigator published Poincare formula of real surfaces and submanifolds of codimension 2. The calculation of this result is obtained by the use of an integral on a Lie group and some symmetric pairsThe head investigator showed that an integral on a Lie group by the use of some symmetric pairs is effective in formulation of Poincare formulae in the other homogeneous spaces. Takahashi, Kang, Sakai and the head investigator has studied integral geometry of almost complex submanifolds in homogeneous almost Hermitian spaces and formulated Poincare formulae of almost complex submanifolds in homogeneous almost Hermitian spaces, which are generalization of classical and fundamental formulae in complex projective spaces obtaind by Santalo. Sakai has generalized these results