On the Weyl conformal invariance on manifolds with various geometric structures and its vanishing of the invariant

各种几何结构流形上的Weyl共形不变性及其不变量的消失

• 批准号: 12640082
• 负责人: KAMISHIMA Yoshinobu
• 金额: $ 2.18万
• 依托单位: Department of Mathematics, Tokyo Metropolitan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640082/
• 关键词: QCC-structure; Uniformization; Holonomy group; Developing map; Spherical CR Geometry; C-R curvature tenso; Parabolic geometry; Quaternionic hyperbolic geometry; QCC-strutme; spherical CR 幾何; 展開写像; ホロノミー群; Uniformization; C-M曲率テンサー; 4元数双曲幾何 /

1. Bochner curvature flat locally conformal Kahler manifolds: The analogue of Weyl curvature tensor on Kahler manifolds is called the Bochner curvature tensor. As it is a local invariant, the definition makes sense on locally conformal Kahler manifolds (l.c. K. manifolds). We have classified compact Bochner flat Kahler manifolds several years ago. In this continuation, we classified more generally noncompact Bochner flat l.c. K. manifolds.2. Symmetry and Global rigidity: When there exists a closed noncompact geometric flow, called Lee-Cauchy-Riemann transformations on a compact l.c. K.manifold M, we have shown a rigidity that M will be isometric to the Hopf manifold of standard type.3. Quaternionic Carnot-Caratheodory structure: We introduced a quaternionic C-C structure on (4n+3)-manifold M and constructed a curvature tensor T which is conformal invariant w. r. t. that geometric structure. If the curvature T vanishes, then we proved that M is locally modelled on the spherical pseudo-quaternionic geometry (Aut_<QC>(S^<4n+3>), S^<4n+3>). In particular, we have established the parabolic geometry on the boundary of the compactification of noncompact semisimple symmetric space of rank 1

1. Bochner 曲率平坦局部共形 Kahler 流形：Kahler 流形上 Weyl 曲率张量的类似物称为 Bochner 曲率张量。由于它是局部不变量，因此该定义对于局部共形卡勒流形（l.c. K. 流形）有意义。几年前我们对紧凑型 Bochner 扁平 Kahler 流形进行了分类。在此延续中，我们对非紧凑 Bochner 扁平 l.c. 进行了更一般的分类。 K.流形.2。对称性和全局刚性：当存在一个封闭的非紧几何流时，称为紧 L.C. 上的 Lee-Cauchy-Riemann 变换。 K.流形M，我们已经证明了M与标准型Hopf流形等距的刚度。3．四元Carnot-Caratheodory结构：我们在(4n+3)-流形M上引入了四元C-C结构，并构造了一个共形不变量w的曲率张量T。河t。那个几何结构。如果曲率 T 消失，则我们证明 M 是在球面伪四元数几何上局部建模的 (Aut_<QC>(S^<4n+3>), S^<4n+3>)。特别地，我们在1阶非紧半简单对称空间的紧化边界上建立了抛物线几何

项目成果

Y.Kamishima(神島芳宣): "Geomatric rigidity of Spherical hypersurfaces in quaternionic manifolds"Asian Journal of Mathematics. 3. 519-556 (1999)

Y. Kamishima (Yoshinobu Kamishima)：“四元流形中球面超曲面的几何刚性”亚洲数学杂志 3. 519-556 (1999)。

神島芽宣: "Rigidify of Ohata-Ferrand's type on Compact h on Kahler l.O.K manifolds"数理研講究録. 1223. 69-79 (2001)

Meinobu Kamishima：“Kahler l.O.K 流形上紧凑 h 的 Ohata-Ferrand 类型的刚性化”数学研究报告。1223. 69-79 (2001)。

神島芳宣: "Rigidity of Obata-Ferrand's type on Compact on Kahler L.C.K manifold"数理研講究録. 1223. 69-79 (2001)

Yoshinobu Kamishima：“Kahler L.C.K 流形上紧致型 Obata-Ferrand 类型的刚性”数学研究报告 1223. 69-79 (2001)。

Y.Kamishima(神島芳宣): "Note on locally conformal kahler surfaces"Geometriae Dedicata. 81. 11-11 (2001)

Y.Kamishima（Yoshinobu Kamishima）：“局部共形卡勒曲面的注释”Geometriae Dedicata。 81. 11-11 (2001)

Yoshinobu Kamishima: "Rigidity of Obata-Ferrand's type on compact non-Kahler l.c.K.manifolds"Lecture notes in MSRI. 1223. 69-79 (2001)

Yoshinobu Kamishima：“Obata-Ferrand 类型在紧凑型非 Kahler l.c.K. 流形上的刚性”MSRI 中的讲义。