Projective contact geometry and singularity theory

射影接触几何和奇点理论

• 批准号: 10440013
• 负责人: ISHIKAWA Goo
• 金额: $ 8.45万
• 依托单位: HOKKAIDOUNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440013/
• 关键词: simple singularity; symplectic defect; contact geometry; Grassmannian geometry; Monge-Ampere equation; Gauss mapping; dual variety; developable submanifold; シンプレクティック・モデュライ; 接触同値; ギベンタル予想; 非ホロノーム横断性定理; 非ホロノームホモトピー原理; 可展面; 等径超曲面; 極小曲面; ルジャン

We have organized the joint works: The head investigator and the investigator Toru Morimoto (see the references); the head investigator and the investigators Reiko Miyabka and Makoto Kimura (submitted); the head Investigator and Ilya Bogaevski(see the references); the head investigator and S. Janeczko (submitted); the head investigator and V. Zakalyukin; and several other projects with investigators. For instance, we have the following result: The isotropic bifurcation problem is reduced to the classification of varieties by symplectomorphisms in the reduced space. The complete symplectic classification of Bruce-Gaffhey's plane curve singularites is provided and is applied to obtain naturally the Lagrangian openings. Moreover we study the Monge-Ampere equations from the view points of contact geometry and investigate the global structure and singularities of Monge-Ampere equations. Also we have the results on the singularities of Gauss mappings and dual varieties, appearing in the com formal geometry and Grassmannian geometry and application to differential equations. We construct developable submanifolds by using minimal submanifolds, harmonic mappings and calibrations.

我们已经组织了联合作品：首席研究员和调查员Toru Morimoto（请参阅参考文献）；调查员和调查人员宫本Reiko和Makoto Kimura（提交）；首席调查员和Ilya Bogaevski（请参阅参考文献）；首席调查员和S. Janeczko（已提交）；首席调查员和V. Zakalyukin；以及其他一些与调查人员的项目。例如，我们有以下结果：各向同性分叉问题通过减少空间中的符号晶状体形态降低为品种的分类。提供了Bruce-Gaffhey平面曲线奇异物的完整符号分类，并应用于自然地获得Lagrangian开口。此外，我们从接触几何学的观点研究了Monge-Ampere方程，并研究了Monge-Ampere方程的全球结构和奇异性。同样，我们也有关于高斯映射和双重品种的奇异性的结果，该品种出现在com形式的几何形状和格拉曼尼亚的几何形状以及对微分方程的应用中。我们通过使用最小的亚策略，谐波映射和校准来构建可开发的亚体。

项目成果

I.Bogaevskii: "Lagrange Mappings of the first open Whitney umbrella"Pacific Journal of Mathematics. (発表予定). (2002)

I.Bogaevskii：“第一张惠特尼伞的拉格朗日映射”《太平洋数学杂志》（即将出版）。

Fukui, T.: "Constructing blow-walytic : isomorphisms"Ann, Inst, Fourier. 51-4. 1071-1087 (2001)

Fukui, T.：“构造blow-walytic：同构”Ann，Inst，Fourier。

石川剛郎: "代数曲線と特異点"共立出版. 363 (2001)

石川武夫：“代数曲线和奇点”Kyoritsu Shuppan 363 (2001)。

S.Izumiya: "The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkouski 3-space"Glasgow.Math.J.. 42. 75-89 (2000)

S.Izumiya：“Minkouski 3-空间中的光锥高斯图和光锥可展开的类空间曲线”Glasgow.Math.J.. 42. 75-89 (2000)

福井敏純: "Modified Nashtriviality of a family of zero-sets of real polynomial mappings" Annales de I'linstitut Fourier. 48-5. 1395-1440 (1998)

Toshizumi Fukui：“实多项式映射的零集的修正纳什平凡性”Annales de Ilinstitut Fourier 48-5 (1998)。