Geometry of curves and surfaces with singularities

具有奇点的曲线和曲面的几何

• 批准号: 19204005
• 负责人: UMEHARA Masaaki
• 金额: $ 14.89万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2010
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-19204005/
• 关键词: 微分幾何; ガウス曲率; 平均曲率; 波面; 特異点; 双対性; 極大曲面; 変曲点; 極小曲面

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.

这项研究的重点是具有奇点的曲线和曲面的几何形状。我们给出了超曲面上 A_k 奇异点的有用准则，并将其应用于超曲面拐点的研究。这个公式使我们能够在不假设周围空间存在的情况下定义波前的 A_k 个奇点。事实上，我们定义了“相干切线丛”的概念，给出了波前和其他几个应用的内在公式。此外，我们研究了Lorentz-Minkowski空间中的最大曲面和德西特空间中的常平均曲率曲面，并构造了几个有趣的新例子，这些例子具有奇点但仍然具有一定的完整性。此外，首席研究员和共同研究员举办了多次研讨会（都是国内的）和国际），与研究相关领域的几何学家进行了许多富有成效的讨论。

项目成果

Singularities of maximal surfaces

• DOI: 10.1007/s00209-007-0250-0
• 发表时间: 2005-10
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: S. Fujimori;K. Saji;M. Umehara;Kotaro Yamada

The geometry of fronts

锋面的几何形状

• 发表时间: 2009
• 期刊: Ann. of Math. 169
• 作者: M. Umehara K. Saji;K. Yamada
• 通讯作者: K. Yamada

Mathematical Proceedings of the Cambridge Philosophical Society

• DOI: 10.1017/s030500411400036x
• 发表时间: 2014-11-01
• 期刊: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
• 影响因子: 0.8
• 作者: Girouard, Alexandre;Parnovski, Leonid;Sher, David A.
• 通讯作者: Sher, David A.

Flat Moebius strips of given isotopy type in R^3 whose centerlines are geodesic or lines of curvature

R^3 中给定同位素类型的平坦莫比斯带，其中心线是测地线或曲率线

• 发表时间: 2008
• 期刊: Geometriae Dedicata 134
• 作者: Y. Kurono;M. Umehara
• 通讯作者: M. Umehara

Flat surfaces with singularities in Euclidean 3-space

• DOI: 10.4310/jdg/1246888486
• 发表时间: 2006-05
• 期刊: Journal of Differential Geometry
• 影响因子: 2.5
• 作者: Satoko Murata;M. Umehara