Homogeneous spaces and variational problems

齐次空间和变分问题

• 批准号: 12640058
• 负责人: TASAKI Hiroyuki
• 金额: $ 2.24万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640058/
• 关键词: homogeneous space; symmetric space; variational problem; integral geometry; Kahler angle; 極小曲面

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.

首席研究员开发了庞加莱积分公式的显式表达式，以便将这些积分几何公式应用于齐次空间中的变分问题。他获得了复射影空间中真实表面的庞加莱公式用这些表面的卡勒角的明确表示。这是第一个明确的，其中交集数的积分不由子流形体积的乘积来表示。此后，为了将该公式推广到复射影空间中的一般实子流形，他定义了多个卡勒角，这是卡勒角的推广。根据多重卡勒角，他发展了复射影空间中一般实子流形的庞加莱公式。因此，可以获得多个卡勒角的一些积分与子流形体积之间的关系，并将成为解决变分问题的工具。

项目成果

D.Hirohashi, H.Song, R.Takagi, H.Tasaki: "Minimal orbits of the isotropy groups of symmetric spaces of compact type"Diferential Geometry Appl.. 13. 167-177 (2000)

D.Hirohashi、H.Song、R.Takagi、H.Tasaki：“紧型对称空间各向同性群的最小轨道”微分几何应用.. 13. 167-177 (2000)

H.Tasaki: "Generalization of Kahler angle and integral geometry in complex projective spaces"Steps in Differential Geometry. 349-361 (2001)

H.Tasaki：“复杂射影空间中卡勒角和积分几何的推广”微分几何的步骤。

H. Kaji, O. Yasukura: "Tangent loci and certain linear sections of adjoint varieties"Nagoya Math. J.. 158. 63-72 (2000)

H. Kaji，O. Yasukura：“伴随变体的切线轨迹和某些线性部分”名古屋数学。

O.Ikawa, T.Sakai, H.Tasaki: "Orbits of Hermann actions"Osaka J. Math.. 38. 923-930 (2000)

O.Ikawa、T.Sakai、H.Tasaki：“赫尔曼作用的轨道”Osaka J. Math.. 38. 923-930 (2000)

R. Aiyama, K. Akutagawa: "Kenmotsu type representation formula, for surfaces with prescribed mean curvature in the hyperbolic 3-space"J. Math. Soc. Japan. 52. 877-898 (2000)

R. Aiyama，K. Akutakawa：“Kenmotsu 型表示公式，用于双曲 3 空间中具有规定平均曲率的表面”J。