Study on the geometry of symmetric spaces and their totally geodesic submanifolds

对称空间及其全测地子流形的几何研究

• 批准号: 13640077
• 负责人: NAITOH Hiroo
• 金额: $ 2.11万
• 依托单位: Yamaguchi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640077/
• 关键词: symmetric space; symmetric submanifold; totally geodesic submanifold; Grassmann geometry; Lie group; Lie algebra; Jordan algebra; symmetric R-space; 部分多様体; 左不変計量; ジョルダン三項対; コンパクトLie群

This investigation is on totally geodesic submanifolds of Riemannian symmetric spaces and the Grassmann geometry of submanifolds associated with them. Such typical submanifolds are symmetric submanifolds.1.Fundamental results on symmetric submanifolds(1)We clarified the relationship between the construction of symmetric submanifolds and the theory of Jordan triple system and the associated symmetric R-space, and obtained a summary on the history and transition on these research fields.(2)We next clarified the details of symmetric submanifolds in the higher-rank irreducible Riemannian symmetric spaces of noncompact type. This is a collaboration with Berndt, Eschenburg, and Tsukada.(3)Summing up these results, we published a paper on the classification of symmetric submanifolds of general Riemannian symmetric spaces in Japanese. This is a collaboration with Tsukada. This result was announced in a JSPS-DFG seminar held at Kyoto University. A translation of this paper will be also issued in the journal "Sugaku Expositions" of the American Mathematical Society.2.Development into another Grassmann geometryAs a development of this research, we understood the study the Grassmann geometries on Lie groups with left invariant metric. So we studied the Grassmann geometries on the 3-dimensional nilpotent Lie group called Heisenberg group and two 3-dimensional unimodular Lie groups, and obtained the classification of their Grassmann geometries and the details about the associated surface theories. As a result, we found that the Grassmann geometry is closely related to the structure of Lie group. The study for Heisenberg case is a collaboration with Inoguchi and Kuwabara.3.A view for future studyA problem remaining in this research is the complete classification of general totally geodesic submanifolds of Riemannian symmetric spaces. Aso, the Grassmann geometry on Lie groups should be developed much.

本文研究黎曼对称空间的全测地子流形以及与之相关的子流形的格拉斯曼几何。1.关于对称子流形的基本结果（1）阐明了对称子流形的构造与Jordan三系理论及相应的对称R-空间之间的关系，总结了这些研究领域的历史和过渡。(2)We其次阐明了非紧型高阶不可约黎曼对称空间中对称子流形的细节。这是与Berndt，Bohenburg和Tsukada的合作。（3）总结这些结果，我们在日本发表了关于一般黎曼对称空间的对称子流形的分类的论文。这是与Tsukada的合作。这一结果是在京都大学举行的JSPS-DFG研讨会上宣布的。本文的译文也将发表在美国数学学会的“Sugaku Expositions”杂志上。2.发展到另一种格拉斯曼几何作为本研究的发展，我们理解了李群上具有左不变度量的格拉斯曼几何的研究。因此，我们研究了三维幂零李群Heisenberg群和两个三维幺模李群上的Grassmann几何，得到了它们的Grassmann几何的分类和相应的曲面理论的细节。结果发现，格拉斯曼几何与李群的结构密切相关。Heisenberg情形的研究是与Inoguchi和Kuwabara的合作。3.对未来研究的展望本研究的一个问题是黎曼对称空间的一般全测地子流形的完全分类。阿索认为李群上的Grassmann几何也应该得到进一步的发展。

项目成果

Classification of symmetric submanifolds of symmetric spaces (in Japanese)

对称空间的对称子流形的分类（日语）

• 发表时间: 2003
• 期刊: Sugaku (ed.Math.Soc.of Japan) 55-3
• 作者: Kazumi Tsukada;Hiroo Naitoh
• 通讯作者: Hiroo Naitoh

Hiroo Naitoh: "Symmetric submanifolds and Jordan triple systems"Sophia Kokyuroku in Mathematics, Sophia University. 45. 21-38 (2002)

Hiroo Naitoh：“对称子流形和乔丹三重系统”Sophia Kokyuroku，上智大学数学系。

Symmetric submanifolds and symmetric R-spaces (in Japanese)

对称子流形和对称 R 空间（日语）

• 发表时间: 2002
• 期刊: Lecture Notes Series in Mathematics (osaka University) 7
• 作者: Kazumi Tsukada;Hiroo Naitoh;Hiroo Naitoh
• 通讯作者: Hiroo Naitoh

内藤 博夫: "対称部分多様体と対称R-空間(竹内勝先生メモリアル研究会,小林亮一編)"Lecture Note Series in Mathematics, Osaka University. 7. 195-219 (2002)

Hiroo Naito：“对称子流形和对称 R 空间（Masaru Takeuchi Memorial Study Group，Ryoichi Kobayashi ed.）”数学讲义系列，大阪大学 7. 195-219 (2002)。

Symmetric submanifolds associated with the irreducible symmetric R-spaces

与不可约对称 R 空间相关的对称子流形

• 发表时间: 2005
• 期刊: Mathematische Annalen 332
• 作者: Kazumi;Tsukada;Hiroo;Naitoh;Kazumi Tsukada;Isao Kiuchi;Yoshifumi Ando;Jurgen Berndt
• 通讯作者: Jurgen Berndt