Homogeneous spaces and variational problems

首席研究员介绍了“多重Kahler角”的概念，并表明我们可以通过使用多重Kahler角来明确地描述复射影空间中子流形的积分几何。在复射影平面的情况下，他用康氏得到了更详细的庞加莱公式.这些Poincare公式在真实的曲面的面积估计和Kahler角的积分中有应用。通过这种估计，我们可以得到一个特定的变分问题的最小解。此外，首席研究员发表庞加莱公式的真实的表面和子流形的余维2。这个结果的计算是利用李群上的一个积分和一些对称对得到的。研究者指出，利用一些对称对在李群上的积分在其它齐性空间中的Poincare公式的公式化中是有效的。Takahashi、Kang、Sakai和首席研究员研究了齐次几乎埃尔米特空间中几乎复子流形的积分几何，并推导出齐次几乎埃尔米特空间中几乎复子流形的庞加莱公式，这些公式是Santalo在复射影空间中获得的经典基本公式的推广。Sakai推广了这些结果

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