Homogeneous spaces and variational problems

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The head investigator has developed explicit expressions of Poincare integral formulas in order to apply these formulas of integral geometry to variational problems in homogeneous spaces. He has obtained an explicit representa tion of Poincare formula of real surfaces in the complex projective spaces in terms of the Kahler angles of those surfaces. This is the first explicit one in which the integral of intersection numbers is not expressed by the product of the volumes of submanifolds. After this in order to generalize this formula to those for general real submanifolds in the complex projective spaces he defined multiple Kahler angles which were generalizations of Kahler angle. According to the multiple Kahler angle he has developed Poincare formulas of general real submanifolds in the complex projective spaces. As a conse quence a relation among some integrals of the multiple Kahler angles and the volumes of submanifolds can be obtained and will become a tool for variational problems.