| Project/Area Number |
19540084
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Saga University |
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Principal Investigator |
MAEDA Sadahiro Saga University, 理工学部, 教授 (40181581)
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| Co-Investigator(Kenkyū-buntansha) |
ICHIKAWA Takashi 佐賀大学, 理工学部, 教授 (20201923)
SEI Keimei 佐賀大学, 理工学部, 教授 (50274577)
HIROSE Susumu 佐賀大学, 理工学部, 准教授 (10264144)
古用 哲夫 島根大学, 総合理工学部, 教授 (40039128)
足立 俊明 名古屋工業大学, 工学部, 教授 (60191855)
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| Project Period (FY) |
2007 – 2009
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| Project Status |
Completed (Fiscal Year 2009)
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| Budget Amount *help |
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
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| Keywords | 微分幾何学 / 部分多様体 / 曲線論 / 実超曲面論 / 極小等質線織実超曲面 / 複素双曲型空間 / 複素射影空間 / 特性ベクトル場 / ホロサイクル / 測地球面 / 等質曲線 / 円 / $A_2$型超曲面 / extrinsic geodesics / 等質線織実超曲面 / 実超曲面 / 測地線 / 線織等質実超曲面 / A 型超曲面 / ケーラー多様体 / シューアの定理 |
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| Research Abstract |
It is not too much to say that Riemannian geometry has been developed with the investigation of geodesics. Among many smooth curves on a Riemannian manifold geometers have mainly studied geodesics. In this research, we propose to study some families of "nice" curves containing geodesics in order to investigate some other properties of Riemannian manifolds. It is known that on an arbitrary Riemannian symmetric space every geodesic is an orbit of some one-parameter subgroup of its isometry group. Noticing this fact, we say that a curve on a Riemannian manifold $M$ is a Killing helix if it is an orbit of some one-parameter subgroup of the isometry group of $M$, and we shed some light on the geometric study of them. Since they are integral curves of some Killing vector field, needless to say they are simple ; namely, they do not have self intersection points.
Our program is to pick up some of the Killing helices in connection with some other geometric objects and study them or study other geometric objects by use of their properties. In this research we study Killing helices in connection with submanifolds. On many homogeneous submanifolds in a symmetric space of rank one, some kinds of geodesics are Killing helices if we consider them as curves on a symmetric space of rank one. This suggests to us that Killing helices are related to submanifolds when we study symmetric spaces of rank one.
In the first half of this research, we obtain properties of Killing helices on a symmetric space of rank one which are obtained from the viewpoint of submanifolds. In the latter half, changing the point of view, we obtain properties of submanifolds obtained by making use of some properties of Killing helices on them.
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