2009 Fiscal Year Final Research Report
Differential Geometry and Partial Differential Equations as an application of Singularity theory
| Project/Area Number |
18340013
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Hokkaido University |
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Principal Investigator |
IZUMIYA Shyuichi Hokkaido University, 大学院・理学研究院, 教授 (80127422)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIKAWA Goo 北海道大学, 大学院・理学研究院, 教授 (50176161)
ONO Kaoru 北海道大学, 大学院・理学研究院, 教授 (20204232)
YAMAGUCHI Keizo 北海道大学, 大学院・理学研究院, 教授 (00113639)
OHMOTO Toru 北海道大学, 大学院・理学研究院, 准教授 (20264400)
TONEGAWA Yoshihiro 北海道大学, 大学院・理学研究院, 教授 (80296748)
UMEHARA Masaaki 大阪大学, 大学院・理学研究院, 教授 (90193945)
KOIKE Shigeaki 埼玉大学, 大学院・理工学研究科, 教授 (90205295)
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| Co-Investigator(Renkei-kenkyūsha) |
UMEHAPR Masaaki 大阪大学, 大学院・理学研究科, 教授 (90193945)
KOIKE Shigeaki 埼玉大学, 大学院・理学研究科, 教授 (90205295)
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| Project Period (FY) |
2006 – 2009
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| Keywords | 特異点論 / 微分幾何 / 偏微分方程式 / シンプレクテイック幾何 / 接触幾何 / 画像処理,ブレーン宇宙 / 重力場レンズ / ラグランジュ特異点 / ルジャンドル特異点 |
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| Research Abstract |
In this research project, we applied Singularity Theory to some areas in Mathematics such as Differential Geometry, Symplectic Geometry, non-linear Partial Differential Equations etc, so that we have obtained several results. Moreover, we have obtained related results on some boundary areas such as Astrophysics etc. Especially, we applied Singularity Theory to Differential Geometry of submanifolds in several kinds of space forms. Then we constructed new geometries (Horospherical Geometry, Slant Geometry) and induced some new invariants. We also clarified the geometric meanings of these invariants. As a result, we have a geometric characterization of the singularities andthe shape of event horizons
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