Homogeneous spaces and variational problems

齐次空间和变分问题

• 批准号: 14540058
• 负责人: TASAKI Hiroyuki
• 金额: $ 2.37万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540058/
• 关键词: homogeneous spaces; variational problems; integral geometry; 交叉積分公式; Poincareの公式; Croftonの公式; Kahler角度

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

主要研究人员引入了多重Kahler角的概念，并证明了利用多重Kahler角可以显式刻画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Kang得到了更详细的Poincare公式。这些庞加莱公式在估计实曲面的面积和Kahler角积分方面有应用。通过这个估计，我们可以得到某一变分问题的极小解。此外，负责人还发表了余维为2的实曲面和子流形的Poincare公式。这一结果的计算是通过在李群和一些对称对上的积分得到的。Takahashi，Kang，Sakai和首席研究员研究了齐次几乎厄米特空间中的几乎复子流形的积分几何，建立了齐次几乎厄米特空间中的几乎复子流形的Poincare公式，推广了Santalo在复射影空间中得到的经典公式和基本公式。酒井先生推广了这些结果。

项目成果

H.Tasaki: "Generalization of Kahler angle and integral geometry in complex projective spaces II"Math.Nachr.. 252. 106-112 (2003)

H.Tasaki：“复射影空间中卡勒角和积分几何的推广 II”Math.Nachr.. 252. 106-112 (2003)

M.Itoh: "Contact metric 5-manifolds, CR twistor spaces and integrability"Jour. Math. Phys.. 43・7. 3783-3797 (2002)

M.Itoh：“接触度量 5 流形、CR 扭转空间和可积性”数学杂志 43・7（2002）。

H.Tasaki: "Integral geometry under the action of the first symplectic group"Archiv der Mathematik (Basel). 80. 106-112 (2003)

H.Tasaki：“第一辛群作用下的积分几何”Archiv der Mathematik（巴塞尔）。

O.Ikawa: "Hamiltonian dynamics of a charged particle"Hokkaido Math.J.. 32・3. 661-671 (2003)

大井川：“带电粒子的哈密尔顿动力学”Hokkaido Math.J. 32・3（2003）

H.Tasaki, Kang: "Integral geometry of real surfaces in the complex projective plane"Geometriae Dedicata. 90. 99-106 (2002)

H.Tasaki, Kang：“复射影平面中真实表面的积分几何”Geometriae Dedicata。