2019 Fiscal Year Final Research Report
Combinatorial analysis of proper actions on pseudo-Riemannian symmetric spaces
| Project/Area Number |
16K17594
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Hiroshima University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | Lie 群 / 対称空間 / 不連続群 / 固有な作用 |
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| Outline of Final Research Achievements |
In this research, we have studied proper actions and discontinuous groups for symmetric spaces. The most valuable result in this research is the following: we found deep relationships between the study of proper actions on pseudo-Riemannian symmetric spaces and that of conjugacy classes of totally geodesic submanifolds in non-compact Riemannian symmetric spaces. Totally submanifolds in symmetric spaces can be understood by Lie algebras and root systems (a kind of combinatorics objects) in some sense. In particular, we give a definition of ``Dynkin indices'' of totally geodesic submanifolds in symmetric spaces in terms of sectional curvatures and applied it to the study of proper actions on pseudo-Riemannian symmetric spaces.
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| Free Research Field |
微分幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
本研究のテーマである対称空間上の固有な群作用, 不連続群は, 微分幾何学における主要な研究分野の一つである. 本研究の成果により, 特に擬リーマン対称空間上の不連続群という取扱いの難しい現象が, リーマン対称空間の全測地的部分多様体(``平面内の直線''や``空間内の平面''などの一般化)と呼ばれる基本的な対象の研究と深く関連することが分かった. これはこの研究分野における重要な知見であると思われる.
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