1999 Fiscal Year Final Research Report Summary
Gemetric Structures on Manifolds and Global Analysis
| Project/Area Number |
09440034
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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 | Kanazawa University (1998-1999)
Nara Women's University (1997) |
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Principal Investigator |
KOBAYASHI Osamu Kanazawa University, Faculty of Science, Professor, 理学部, 教授 (10153595)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIOKA Atsushi Kanazawa University, Faculty of Science, Instructor, 理学部, 助手 (30293335)
KITAHARA Haruo Kanazawa University, Faculty of Science, Professor, 理学部, 教授 (60007119)
KODAMA Akio Kanazawa University, Faculty of Science, Professor, 理学部, 教授 (20111320)
KATO Shin Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (10243354)
KATAGIRI Minyo Nara Women's University, Faculty of Science, Associate Professor, 理学部, 助教授 (60263422)
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| Project Period (FY) |
1997 – 1999
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| Keywords | confomal structure / conformal connection / Moebius geometry / projective structure / scalar curvature / Ricci curvature / Schwarzian / vertex |
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| Research Abstract |
Among many geometric structures of a manifold we are mainly interested in those structures which are closely related to the conformal geometry. Here are some of main results of this research project :
1. The scalar curvature equation. This equation describes the scalar curvature under a conformal change of a Riemannian metric. A systematic analysis has been done on non-compact manifolds, and the space of complete confomal metrics with prescribed scalar curvature is made clearer.
2. The Weyl structure. This is a torsion free affine connection that is compatible with a given conformal class. It is shown that the Ricci curvature is a complete invariant of a Weyl structure. Also conformally flat Einstein-Weyl structures on compact manifolds are classified.
3. Moebius geometry. The minimum number of vertices of a regular closed curve on the sphere with given topological type is completely determined in the case when the curve has at most five self-inter-sections. Also we introduce a Schwarzian derivative of a regular curve. This leads to new proofs of injectivity results of Nehari type. A gist is that a confomal strucutre of a manifold induces an integrable projective structure of a regular curve on the manifold. It is shown that injectivity of the projective development map of the curve implies the injectivity of the immersion to Moebius spaces.
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