2010 Fiscal Year Final Research Report
Geometry of curves and surfaces with singularities
| Project/Area Number |
19204005
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | Osaka University |
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Principal Investigator |
UMEHARA Masaaki Osaka University, 大学院・理学研究科, 教授 (90193945)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Kotaro 東京工業大学, 大学院・理工学研究科, 教授 (10221657)
HASHIMOTO Hideya 名城大学, 理工学部, 教授 (60218419)
MASHIMO Katsuya 法政大学, 理工学部, 教授 (50157187)
MABUCHI Toshiki 大阪大学, 大学院・理学研究科, 教授 (80116102)
KOISO Norihito 大阪大学, 大学院・理学研究科, 教授 (70116028)
GOTO Ryushi 大阪大学, 大学院・理学研究科, 准教授 (30252571)
ENOKI Ichiro 大阪大学, 大学院・理学研究科, 准教授 (20146806)
W.F. Rossman 神戸大学, 理学部, 教授 (50284485)
MIYAOKA Reiko 東北大学, 大学院・理学研究科, 教授 (70108182)
KOKUBU Masatoshi 東京電機大学, 工学部, 教授 (50287439)
FUJIMORI Shoichi 福岡教育大学, 教育学部, 准教授 (00452706)
AGAOKA Yoshio 広島大学, 大学院・理学研究科, 教授 (50192894)
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| Project Period (FY) |
2007 – 2010
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| Keywords | 微分幾何 / ガウス曲率 / 平均曲率 / 波面 / 特異点 |
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| Research Abstract |
This research was focused on the geometry of curves and surfaces with singularities. We gave a useful criterion for A_k singular points on hypersurfaces, and applied it to the study of inflection points on hypersurfaces. This riterion enabled us to define A_k singularities of wave front without assuming the existence of an ambient space. In fact, we defined the notion "coherent tangent bundle", giving an intrinsic formulation for wave fronts and several other applications. Moreover, we investigated maximal surfaces in Lorentz-Minkowski space and constant mean curvature surfaces in de Sitter space, and constructed several interesting new examples with singularities but still having certain kind of completeness.
Additionally, the head investigator and coinvestigators held several workshops (both domestic and international), which related in many fruitful discussions with geometers studying relating fields.
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