Homogeneous spaces and variational problems

为了将积分几何中的庞加莱积分公式应用于齐次空间中的变分问题，研究人员开发了庞加莱积分公式的显式表达式。他得到了复射影空间中实数曲面的庞加莱公式用这些曲面的Kahler角的显式表示。这是第一个明确的交集数的积分不是由子流形的体积积表示的。之后为了将这个公式推广到复射影空间中一般实子流形他定义了多个Kahler角这是Kahler角的推广。根据多重Kahler角，建立了复射影空间中一般实子流形的庞加莱公式。由此可以得到多个Kahler角的某些积分与子流形体积之间的关系，这将成为解决变分问题的一个工具。

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.