2013 Fiscal Year Final Research Report
Weierstrass-type representation formulas and their application to surfaces with singularities
| Project/Area Number |
21340016
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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 | Tokyo Institute of Technology |
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Principal Investigator |
YAMADA Kotaro 東京工業大学, 理工学研究科, 教授 (10221657)
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| Co-Investigator(Renkei-kenkyūsha) |
UMEHARA Masaaki 大阪大学, 大学院・理学研究科, 教授 (90193945)
UMEHARA Masaaki 東京工業大学, 大学院・情報理工学研究科, 教授 (90193945)
WAYNE Rossman 神戸大学, 理学部, 教授 (50284485)
YOSHIDA Masaaki 九州大学, 大学院・数理学研究院, 教授 (30030787)
KUROSE Takashi 福岡大学, 理学部, 教授 (30215107)
KUROSE Takashi 関西学院大学, 理学部, 教授 (30215107)
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, 講師 (00452706)
FUJIMORI Shoichi 岡山大学, 理学部, 准教授 (00452706)
KAWAKAMI Yu 九州大学, 大学院・数理学研究院, 助教 (60532356)
KAWAKAMI Yu 山口大学, 理学部, 講師 (60532356)
HONDA Atsufumi 都城工業高等専門学校, 講師
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| Project Period (FY) |
2009-04-01 – 2014-03-31
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| Keywords | 曲面 / 特異点 / フロント / 特異曲率 / 連接接束 / ワイエルストラス表現 |
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| Research Abstract |
We investigated, Weierstrass-type representation formula, global properties of several classes of surfaces with singularities, such as flat surfaces in hyperbolic 3-space, maximal surfaces in Minkowski 3-space, CMC-1 surfaces in de Sitter 3-space, and improper affine sphere in affine 3-space, and obtained a charctreization of completeness, Osserman-type inequalis etc.
In addition, flat trinoids in hyperbolic space and CMC-1 2-noids in de Sitter 3-space are classified. On the other hand, as a basic tool of differential geometry of wave front, we introduced a notion of "sinular curvature" and investigated a rdelationship between singular curvature and behavior of Gaussian curvature. As a result, we obtained Gauss-Bonnet type formula for wave fronts. Moreover, as an intrinsic formulation of wave fronts, we introduced a notion of "coherent tangent bundles" and gave an application of their duality.
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